Handbook of Algebra

HANDBOOK OF ALGEBRA Volume 6 edited by M. HAZEWINKEL CWI, Amsterdam AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK ...

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HANDBOOK OF ALGEBRA Volume 6

edited by M. HAZEWINKEL CWI, Amsterdam

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO North-Holland is an imprint of Elsevier

North-Holland is an imprint of Elsevier 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA Linacre House, Jordan Hill, Oxford OX2 8DP, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2009 Copyright © 2009 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) (0) 1865 843830, fax: (+44) (0) 1865 853333, e-mail: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier. com/locate/permissions, and selecting Obtaining permission to use Elsevier material. Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-444-53257-2 ISSN: 1570-7954

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Preface Basic philosophy Algebra, as we know it today (2008), consists of a great many ideas, concepts, and results, and this was also the case in 1995 when this handbook started (which does not mean that nothing has happened in those 12 years; on the contrary, the ﬁeld of algebra and its applications has developed at a furious pace). A reasonable estimate of the number of all the various different concepts, ideas, deﬁnitions, constructions, results, . . . in algebra would be somewhere between 50 000 and 200 000. Many of these have been named and many more could (and perhaps should) have a “name” or other convenient designation. Even a nonspecialist is quite likely to encounter most of these, either somewhere in the published literature in the form of an idea, deﬁnition, theorem, algorithm. . . or to hear about them, often in somewhat vague terms, and to feel the need for more information. In such a case, if the concept relates to algebra, then one should be able to ﬁnd something in this handbook, at least enough to judge whether it is worth the trouble to try to ﬁnd out more. In addition to the primary information, the numerous references to important articles, books, or lecture notes should help the reader ﬁnd out as much as desired. As a further tool, the index is perhaps more extensive than usual, and is deﬁnitely not limited to deﬁnitions, (famous) named theorems, and the like. For the purposes of this handbook, “algebra” is more or less deﬁned as the union of the following areas of the Mathematics Subject Classiﬁcation Scheme: – 20 (Group theory) – 19 (K-theory this will be treated at an intermediate level; a separate Handbook of K-Theory, which goes into far more detail than the section planned for this Handbook of Algebra is under consideration) – 18 (Category theory and homological algebra, including some of the uses of categories in computer science, often classiﬁed somewhere in Section 68) – 17 (Nonassociative rings and algebras, especially Lie algebras) – 16 (Associative rings and algebras) – 15 (Linear and multilinear algebra. Matrix theory) – 13 (Commutative rings and algebras; here, there is a ﬁne line to tread between commutative algebras and algebraic geometry; algebraic geometry is deﬁnitely not a topic that will be dealt within any detail in this handbook; one day, there will, hopefully, be a separate handbook on that topic) – 12 (Field theory and polynomials) v

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– 11 [Number theory, the part that also used to be classiﬁed under 12 (Algebraic number theory)] – 08 (General algebraic systems) – 06 (Order, lattices, ordered algebraic structures; certain parts; but not topics speciﬁc to Boolean algebras as there is a separate three-volume Handbook of Boolean Algebras) Planning Originally (1992), we expected to cover the whole ﬁeld in a systematic way. Volume 1 would be devoted to what is now called Section 1 (see below), Volume 2 to Section 2, and so on. A quite detailed and comprehensive plan was made in terms of topics that needed to be covered and authors to be invited. That turned out to be an inefﬁcient approach. Different authors have different priorities, and to wait for the last contribution to a volume, as planned originally, would have resulted in long delays. Instead, there is now a dynamic evolving plan. This also makes it possible to take new developments into account. Chapters are still by invitation only according to the then current version of the plan, but the various chapters are published as they arrive, allowing for faster publication. Thus, in this volume 6 of the Handbook of Algebra, the reader will ﬁnd contributions from ﬁve sections. As the plan is dynamic, suggestions from users, both as to topics that could or should be covered, and authors are most welcome and will be given serious consideration by the board and editor. The list of sections looks as follows: Section 1: Linear algebra. Fields. Algebraic number theory Section 2: Category theory. Homological and homotopical algebra. Methods from logic (algebraic model theory) Section 3: Commutative and associative rings and algebras Section 4: Other algebraic structures. Nonassociative rings and algebras. Commutative and associative rings and algebras with extra structure Section 5: Groups and semigroups Section 6: Representations and invariant theory Section 7: Machine computation. Algorithms. Tables Section 8: Applied algebra Section 9: History of algebra For the detailed plan (2008 version), the reader is referred to the Outline of the Series following this preface. The individual chapters It is not the intention that the handbook as a whole can also be a substitute undergraduate or even graduate, textbook. Indeed, the treatments of the various topics will

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be too much dense and professional for that. Basically, the level should be graduate and up, and such material as can be found in P.M. Cohn’s three-volume textbook “Algebra” (Wiley) should, as a rule, be assumed known. The most important function of the chapters in this handbook is to provide professional mathematicians working in a different area with a sufﬁciency of information on the topic in question if and when it is needed. Each of the chapters combines some of the features of both a graduate-level textbook and a research-level survey. Not all the ingredients mentioned below will be appropriate in each case, but authors have been asked to include the following: – Introduction (including motivation and historical remarks). – Outline of the chapter. – Basic concepts, deﬁnitions, and results (these may be accompanied by proofs or (usually better) ideas/sketches of the proofs when space permits). – Comments on the relevance of the results, relations to other results, and applications. – Review of the relevant literature; possibly complete with the opinions of the author on recent developments and future directions. – Extensive bibliography (several hundred items will not be exceptional). The present Volume 1 appeared in 1995 (copyright 1996), Volume 2 in 2000, Volume 3 in 2003, Volume 4 in 2005 (copyright 2006), Volume 5 in 2007. Volume 7 is planned for 2009. Thereafter, we aim at one volume every 2 years (or better). The future Of course, ideally, a comprehensive series of books like this should be interactive and have a hypertext structure to make ﬁnding material and navigation through it immediate and intuitive. It should also incorporate the various algorithms in implemented form as well as permit a certain amount of dialog with the reader. Plans for such an interactive, hypertext, CDROM (DVD)-based (or web-based) version certainly exist, but the realization is still a nontrivial number of years in the future. Kvoseliai, July 2008

Michiel Hazewinkel

Kaum nennt man die Dinge beim richtigen Namen so verlieren sie ihren gefährlichen Zauber (You have but to know an object by its proper name for it to lose its dangerous magic) Elias Canetti

Outline of the series (as of July 2008)

Plilosophy and principles of the Handbook of Algebra Compared to the outline in Volume 1, this version differs in several aspects. First, there is a major shift in emphasis away from completeness as far as the more elementary material is concerned and toward more emphasis on recent developments and active areas. Second, the plan is now more dynamic in that there is no longer a ﬁxed list of topics to be covered, determined long in advance. Instead, there is a more ﬂexible nonrigid list that can and does change in response to new developments and availability of authors. The new policy, which started with Volume 2, is to work with a dynamic list of topics that should be covered, to arrange these in sections and larger groups according to the major divisions into which algebra falls, and to publish collections of contributions (i.e. chapters) as they become available from the invited authors. The coding below is by style and is as follows: – Author(s) in bold, followed by chapter title: articles (chapters) that have been received and are published or are being published in this volume. – Chapter title in italic: chapters that are being written. – Chapter title in plain text: topics that should be covered but for which no author has yet been deﬁnitely contracted. Chapters that are included in Volumes 1–6 have a (x; yy pp) after them, where “x” is the volume number and “yy” is the number of pages. Compared to the plan that appeared in Volume 1, the section on “Representation and invariant theory” has been thoroughly revised. Compared with the outline that appeared in Volume 4, section 4H (Rings and algebras with additional structure has been split into two parts: 4H: Hopf algebras and related structures; 4I: Other rings and algebras with additional structure. The old section 4I (Witt vectors) has been absorbed into the section on Hopf algebras. The other changes of this current version compared to the one in Volume 4 (2006) are relatively minor: mostly the addition of a number of topics.

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Compared with the outline in Volume 5, there are only minor changes. Editorial set-up Managing editor: Michiel Hazewinkel Editorial board: M. Artin, M. Nagata, C. Procesi, O. Tausky-Todd† , R.G. Swan, P.M. Cohn, A. Dress, J. Tits, N.J.A. Sloane, C. Faith, S.I.A’dyan, Y. Ihara, L. Small, E. Manes, I.G. Macdonald, M. Marcus, L.A. Bokut’, Eliezer (Louis Halle) Rowen, John S. Wilson, Vlastimil Dlab. Planned publishing schedule (as of September 2007) 1996: Volume 1 (published) 2001: Volume 2 (published) 2003: Volume 3 (published) 2006: Volume 4 (published) 2007 (last quarter): Volume 5 (published) 2008: (last quarter) Volume 6 Further volumes, roughly, at the rate of one every year. Contents of the Handbook of Algebra Section 1: Linear algebra. Fields. Algebraic number theory Section 2: Category theory. Homological and homotopical algebra Section 3: Commutative and associative rings and algebras Section 4: Other algebraic structures. Nonassociative rings and algebras. Commutative and associative rings and algebras with additional structure. Section 5: Groups and semigroups Section 6: Representations and invariant theory Section 7: Machine computation. Algorithms. Tables Section 8: Applied algebra Section 9: History of algebra More detailed plans for Sections 1–9 follow below. Very little in the way of detailed plans has been made so far for Sections 7–9. Experience has shown that often 40–50 pages is a suitable length for a chapter, but both longer and shorter chapters have already occurred and authors are basically free in this matter. It is not the intention that the handbook as a whole will also be a substitute undergraduate textbook. Basically, the level should be graduate and up, and such material as can be found in P.M. Cohn’s three-volume textbook “Algebra” (Wiley) can as a rule be assumed known. A main function of the articles in the handbook is to provide professional mathematicians working in a different area with a sufﬁciency of information on the topic in question if and when he needs it. In the following, there is a more detailed description of a number of sections in various volumes, which have more elementary sounding titles. These are partly intended as quick condensed surveys, establishing basic deﬁnitions and notations among other things but mainly as opportunities to discuss modern developments and new kinds of applications.

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Mostly, each chapter should combine some of the features of a graduate–level textbook and a research-level survey. Not all the ingredients mentioned below will be appropriate in each case, but basically the authors should strive to include the following: – Introduction (including motivation and historical remarks). – Outline of the chapter. – Basic concepts, deﬁnitions, and results [these can be accompanied by proofs or (usually better) ideas/sketches of the proofs if space permits]. – Comments on the relevance of the results, relations to other results, and applications. – Review of the relevant literature; possibly complete with the opinions of the author on recent developments and future directions. – Extensive bibliography (several hundred items will not be exceptional). Often, it will be better to present a result in terms of its central core instead of the widest possible but technically involved generalization; the latter can be mentioned later, for instance in terms of an appropriate reference. Section 1. Linear algebra. Fields. Algebraic number theory A. Linear algebra G.P. Egorychev, Van der Waerden conjecture and applications (1; 22 pp) V.L. Girko, Random matrices (1; 52 pp) Alexander E. Guterman, Matrix invariants over semirings (6; 35 pp) J.A. Hermida-Alonso, Linear algebra over commutative rings (3; 49 pp) A.N. Malyshev, Matrix equations. Factorization of matrices (1; 38 pp) L. Rodman, Matrix functions (1; 38 pp) Correction to the chapter by L. Rodman, Matrix functions (3; 1 p) Kazimierz Szymiczek, Quadratic forms (6; 43 pp) Linear inequalities Orderings (partial and total) on vectors and matrices Structured matrices Matrices over the integers and other (special) rings and ﬁelds Quasideterminants and determinants over noncommutative ﬁelds Nonnegative matrices, positive deﬁnite matrices, and doubly nonnegative matrices Generalized inverses of matrices Matrix invariants over noncommutative rings B. Linear (in)dependence J.P.S. Kung, Matroids (1; 28 pp) C. Algebras arising from vector spaces Clifford algebras Other algebras arising from vector spaces

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D. Fields, Galois theory, and algebraic number theory (There is also a chapter planned on ordered ﬁelds in Section 4I; see Section 4H for a chapter on Picard–Vessiot theory) Toma Albu, From ﬁeld theoretic to abstract co-Galois theory (5; 81 pp) J.K. Deveney, J.N. Mordeson, Higher derivation Galois theory of inseparable ﬁeld extensions (1; 34 pp) I. Fesenko, Complete discrete valuation ﬁelds. Abelian local class ﬁeld theories (1; 48 pp) M. Jarden, Inﬁnite Galois theory (1; 52 pp) R. Lidl, H. Niederreiter, Finite ﬁelds and their applications (1; 44 pp) W. Narkiewicz, Global class ﬁeld theory (1; 30 pp) H. van Tilborg, Finite ﬁelds and error correcting codes (1; 28 pp) Skew ﬁelds and division rings. Brauer group Topological and valued ﬁelds. Valuation theory Zeta and L-functions of ﬁelds and related topics Structure of Galois modules Constructive Galois theory Dessins d’enfants Hopf–Galois theory E. Non-Abelian class ﬁeld theory and the Langlands program (to be arranged in several chapters by Y. Ihara) F. Generalizations of ﬁelds and related objects U. Hebisch, H.J. Weinert, Semirings and semiﬁelds (1; 38 pp) G. Pilz, Near rings and near ﬁelds (1; 36 pp) Section 2. Category theory. Homological and homotopical algebra. Methods from logic A. Category theory S. Mac Lane, I. Moerdijk, Topos theory (1; 28 pp) Ernie Manes, Monads of sets (3; 48 pp) Martin Markl, Operad and PROPs (5; 54 pp) Boris I. Plotkin, Algebra, categories, and databases (2; 71 pp) Peter S. Scott, Some aspects of categories in computer science (2; 71 pp) Ross Street, Categorical structures (1; 50 pp) Algebraic structures in braided categories B. Homological algebra. Cohomology. Cohomological methods in algebra. Homotopical algebra Ronald Brown, Crossed complexes and higher homotopy groupoids as noncommutative tools for higher dimensional local-to-global problems (6; 32 pp)

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Jon F. Carlson, The cohomology of groups (1; 30 pp) A. Generalov, Relative homological algebra. Cohomology of categories, posets, and coalgebras (1; 28 pp) A. Ya Helemskii, Homology for the algebras of analysis (2; 143 pp) J.F. Jardine, Homotopy and homotopical algebra (1; 32 pp) B. Keller, Derived categories and their uses (1; 32 pp) Volodymyr Lyubashenko, Oleksandr Manchuk, A(\infty)-algebras, categories, and -functors (5; 46 pp) Boris V. Novikov, 0-cohomology of semigroups (5; 23 pp) Galois cohomology (H. Koch) Cohomology of commutative and associative algebras Cohomology of Lie algebras Cohomology of group schemes C. Algebraic K-theory Aderemi Kuku, Classical Algebraic K-theory: the functors K0 , K1 , K2 (3; 55 pp) Aderemi Kuku, Algebraic K-theory: the higher K-functors (4; 106 pp) Grothendieck groups K2 and symbols KK-theory and EXT Hilbert C*-modules Index theory for elliptic operators over C* algebras Simplicial algebraic K-theory Chern character in algebraic K-theory Noncommutative differential geometry K-theory of noncommutative rings Algebraic L-theory Cyclic cohomology Asymptotic morphisms and E-theory Hirzebruch formulae D. Model theoretic algebra (see also Paul C. Eklof, Whitehead modules, in Section 3B) Mike Prest, Model theory for algebra (3; 31 pp) Mike Prest, Model theory and modules (3; 34 pp) Frank O. Wagner, Stable groups (2; 36 pp) Logical properties of ﬁelds and applications Recursive algebras Logical properties of Boolean algebras The A-E-K theorem and its relatives E. Rings up to homotopy. Homotopic algebra Rings up to homotopy Simplicial algebras

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Section 3. Commutative and associative rings and algebras A. Commutative rings and algebras (See also Carl Faith, Coherent rings and annihilator conditions in matrix and polynomial rings in Section 3B) J.P. Lafon, Ideals and modules (1; 24 pp) Dorin Popescu, Artin approximation (2; 45 pp) Rafael H. Villareal, Monomial algebras and polyhedral geometry (3; 62 pp) General theory. Radicals, prime ideals, etc. Local rings (general). Finiteness and chain conditions Extensions. Galois theory of rings Modules with quadratic form Homological algebra and commutative rings. Ext, Tor, etc. Special properties (p.i.d., factorial, Gorenstein, Cohen–Macauley, Bezout, Fatou, Japanese, excellent, Ore, Prüfer, Dedekind, . . . and their interrelations) Finite commutative rings and algebras (see also the chapter by A.A. Nechaev in Section 3B) Localization. Local-global theory Rings associated to combinatorial and partial order structures (straightening laws, Hodge algebras, shellability, . . .) Witt rings of quadratic forms and real spectra B. Associative rings and algebras (see also “Freeness theorem for groups, rings, and Lie algebras” in Section 5A) Victor Bavula, Filter dimension (4; 28 pp) Paul M. Cohn, Polynomial and power series rings. Free algebras, ﬁrs, and semiﬁrs (1; 30 pp) Paul C. Eklof, Whitehead modules (3; 23 pp) Edgar E. Enochs, Flat covers (3; 21 pp) Alberto Facchini, The Krull–Schmidt theorem (3; 42 pp) Carl Faith, Coherent rings and annihilation conditions in matrix and polynomial rings (3; 31 pp) V.K. Kharchenko, Simple, prime, and semiprime rings (1; 52 pp) V.K. Kharchenko, Fixed rings and noncommutative invariant theory (2; 27 pp) T.Y. Lam, Hamilton’s quaternions (3; 26 pp) Aleksandr A. Necchaev, Finite rings with applications (5; 108 pp) Sudarshan K. Sehgal, Group rings and algebras (3; 96 pp). Askar A. Tuganbaev, Modules with distributive submodule lattice (2; 25 pp) Askar A. Tuganbaev, Serial and distributive modules and rings (2; 25 pp) Askar A. Tuganbaev, Modules with the exchange property and exchange rings (2; 25 pp) Askar A. Tuganbaev, Max rings and V-rings (3; 23 pp)

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Askar A. Tuganbaev, Semiregular, weakly regular, and πregular rings (3; 25 pp) A. van den Essen, Algebraic microlocalization and modules with regular singularities over ﬁltered rings (1; 28 pp) F. Van Oystaeyen, Separable algebras (2; 66 pp) K. Yamagata, Frobenius rings (1; 48 pp) Classiﬁcation of Artinian algebras and rings General theory of associative rings and algebras Rings of quotients. Noncommutative localization. Torsion theories von Neumann regular rings Lattices of submodules PI rings Generalized identities Endomorphism rings, rings of linear transformations, matrix rings Homological classiﬁcation of (noncommutative) rings Dimension theory Duality. Morita-duality Commutants of differential operators Rings of differential operators Graded and ﬁltered rings and modules (also commutative) Goldie’s theorem, Noetherian rings, and related rings Sheaves in ring theory Koszul algebras Hamiltonian algebras Algebraic asymptotics Antiautomorphisms, especially for simple central algebras C. Coalgebras (See also the chapter by Tomasz Brzezinski in Section 4H) W. Michaelis, Coassociative coalgebras (3; 120 pp) Co-Lie-algebras D. Deformation theory of rings and algebras (including Lie algebras) Yu Khakimdzanov, Varieties of Lie algebras (2; 35 pp) Deformation theory of rings and algebras (general) Deformation theoretic quantization Section 4. Other algebraic structures. Nonassociative rings and algebras. Commutative and associative algebras with extra structure A. Lattices and partially ordered sets David Kruml, Jan Paseka, Algebraic and categorical aspects of quantales (5; 43 pp) Ales Pultr, Frames (3; 63 pp)

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B. Boolean algebras C. Universal algebra and general algebraic concepts and constructions Universal algebra (general) Radicals D. Varieties of algebras, groups . . . (See also Yu Khakimdzanov, Varieties of Lie algebras in Section 3D) V.A. Artamonov, Varieties of algebras (2; 30 pp) V.A. Artamonov, Quasi-varieties (3; 19 pp) Varieties of groups Varieties of semigroups E. Lie algebras (See also “Freeness theorems for groups, rings, and Lie algebras” in Section 5A) Yu A. Bahturin, M.V. Zaitsev, A.A. Mikhailov, Inﬁnite-dimensional Lie superalgebras (2; 30 pp) Michel Goze, Yusupdjan Khakimdjanov, Nilpotent and solvable Lie algebras (2; 51 pp) A.T. Molev, Gel’fand–Tsetlin bases (4; 69 pp). Could also have been placed in Section 6D. Christophe Reutenauer, Free Lie algebras (3; 21 pp) General structure theory Classiﬁcation theory of semisimple Lie algebras over R and C The exceptional Lie algebras Universal enveloping algebras Modular (ss) Lie algebras (including classiﬁcation) Inﬁnite dimensional Lie algebras (general) Kac–Moody Lie algebras Finitary Lie algebras Standard bases Afﬁne Lie algebras and Lie super algebras and their representations Kostka polynomials F. Jordan algebras (ﬁnite and inﬁnite dimensional and including their cohomology theory) G. Other nonassociative algebras (Malcev, alternative, Lie admissible . . .) Mal’tsev algebras Alternative algebras H. Hopf algebras and related structure (see also “Hopf–Galois theory” in Section 1D) (see also “Co-Galois theory” in Section 1D) (see also “Algebraic structures on braided categories” in Section 2A)

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(see also “Representation theory of semisimple Hopf algebras” in Section 6D) Katsutoshi Amano, Akaira Masuoka, Mitsuhiro Takeuchi, Hopf algebraic approach to Picard–Vessiot theory (6; 50 pp) Gabriella Böhm, Hopf algebroids (6; 44 pp) Tomasz Brzezinski, ´ Comodules and corings (6; 72 pp) Mia Cohen a.o., Hopf algebras (general) (4; 87 pp) Michiel Hazewinkel, Witt vectors. Part 1 (6; 148 pp) Jae-Hoon Kwon, Crystal graphs and the combinatorics of Young tableaux (6; 32 pp) Dominiue Manchon, Hopf algebras in renormalization (5; 67 pp) Akira Masuoka, Classiﬁcation of semisimple Hopf algebras (5; 29 pp) A.I. Molev, Yangians and their applications (3; 54 pp) Frédéric Patras, Lambda-rings (3; 34 pp) Classiﬁcation of pointed Hopf algebras Quantum groups (general) Formal groups p-divisible groups Combinatorial Hopf algebras Symmetric functions Special functions and q-special functions, 1–2 variable case Quantum groups and multiparameter q-special functions Noncommutative geometry a` la Connes Noncommutative geometry from the algebraic point of view Noncommutative geometry, categorical point of view Solomon descent algebras Hopf algebras and operads Noncommutative symmetric functions and quasisymmetric functions Witt vectors. Part 2 Trees, dendriform algebras, and dialgebras Quantum differential geometry, quantum calculus, quantum group approach to noncommutative geometry Connes–Baum theory

I. Other rings and algebras with additional structure Alexander Levin, Difference algebra (4; 100 pp) Graded and super algebras (commutative, associative; for Lie superalgebras, see Section 4E) Topological rings Ordered and lattice-ordered rings and algebras. Rings and algebras with involution. C*-algebras Differential algebra Baxter algebras Ordered ﬁelds

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Hypergroups Stratiﬁed algebras Section 5. Groups and semigroups A. Groups (see also “Groups and semigroups of automata transformations” in Section 5B) Laurent Bartholdi, Rostislav I. Grigorchuk, Zoran Sunik, Branch groups (3; 129 pp) Meinolf Geck, Gunter Malle, Reﬂection groups. Coxeter groups (4; 38 pp) A.V. Mikhalev, A.P. Mishina, Inﬁnite Abelian groups: methods and results (2; 48 pp) V.I. Senashov, Groups with ﬁniteness conditions (4; 27 pp) M.C. Tamburini. E.V. semirnov, Hurwitz groups and Hurwitz generation (4; 28 pp) V.V. Vershinin, Braid groups (4; 22 pp) Simple groups, sporadic groups Representations of the ﬁnite simple groups Diagram methods in group theory (Presentations of groups) Abstract (ﬁnite) groups. Structure theory. Special subgroups. Extensions and decompositions Solvable (soluble) groups, nilpotent groups, p-groups. Fitting classes, Schunck classes. Thompson’s list of minimal nonsolvable groups to be included Inﬁnite soluble groups Word and conjugacy problems Burnside problem Combinatorial group theory Distance transitive graphs and their groups Free groups (including actions on trees) Formations Inﬁnite groups. Local properties The inﬁnite dimensional classical groups Inﬁnite simple groups Algebraic groups. The classical groups. Chevalley groups Chevalley groups over rings Other groups of matrices. Discrete subgroups Groups with BN-pair, Tits buildings . . . Groups and (ﬁnite combinatorial) geometry “Additive” group theory Probabilistic techniques and results in group theory Self-similar groups Frobenius group Just-inﬁnite groups Automorphism groups of groups Automorphism groups of algebras and rings

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Freeness theorems (in groups and rings and Lie algebras) Groups with perscribed systems of subgroups Automatic groups Groups with minimality and maximality conditions (School of N.S. Chernikov) Lattice-ordered groups Linearly and totally ordered groups Finitary groups Random groups Hyperbolic groups Inﬁnite dimensional groups B. Semigroups (See also the chapter by Boris Novikov in Section 2B) Semigroup theory. Ideals, radicals, structure theory Semigroups and automata theory and linguistics Groups and semigroups of automata transformations C. Algebraic formal language theory. Combinatorics of words D. Loops, quasigroups, heaps. . . Quasigroups in combinatorics E. Combinatorial group theory and topology (See also: “Diagram methods in group theory” in Section 5A) Section 6. Representation and invariant theory A. Representation theory. General Representation theory of rings, groups, algebras (general) Modular representation theory (general) Representations of Lie groups and Lie algebras. General Multiplicity free representations B. Representation theory of ﬁnite and discrete groups and algebras Representation theory of ﬁnite groups in characteristic zero Modular representation theory of ﬁnite groups. Blocks Representation theory of the symmetric groups (both in characteristic zero and modular) Representation theory of the ﬁnite Chevalley groups (both in characteristic zero and modular) Modular representation theory of Lie algebras C. Representation theory of “continuous groups” (linear algebraic groups, Lie groups, loop groups. . .) and the corresponding algebras

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(see also Gel’fand–Tsetlin bases for classical Lie algebras in Section 4E by A.I. Molev) A.U. Klimyk, Inﬁnite dimensional representations of quantum algebras (2; 26 pp) Representation theory of compact topolgical groups Representation theory of locally compact topological groups Representation theory of SL2(R). . . Representation theory of the classical groups. Classical invariant theory Classical and transcendental invariant theory Reductive groups and their representation theory Unitary representation theory of Lie groups Finite dimensional representation theory of the ss Lie algebras (in characteristic zero); structure theory of semisimple Lie algebras Inﬁnite dimensional representation theory of ss Lie algebras. Verma modules Representation of Lie algebras. Analytic methods Representations of solvable and nilpotent Lie algebras. The Kirillov orbit method Orbit method, Dixmier map. . . for ss Lie algebras Representation theory of the exceptional Lie groups and Lie algebras Representation theory of “classical” quantum groups Duality in representation theory Representation theory of loop groups and higher dimensional analogues, gauge groups, and current algebras Representation theory of Kac–Moody algebras Invariants of nonlinear representations of Lie groups Representation theory of inﬁnite dimensional groups like GL∞ Metaplectic representation theory D. Representation theory of algebras Xueqing Chen, Tomás Pospíchal, Ki-Bong Nam, Quivers and representations (6; 53 pp) Representations of rings and algebras by sections of sheafs Representation theory of algebras (Quivers, Auslander–Reiten sequences, almost split sequences. . .) Tame algebras Ringel–Hall algebras Composition algebras Quasi-hereditary algebras Cellular algebras Representation theory of (semisimple) Hopf algebras E. Abstract and functorial representation theory Serge Bouc, Burnside rings (2; 60 pp)

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P. Webb, A guide to Mackey functors (2; 32 pp) Abstract representation theory F. Representation theory and combinatorics G. Representations of semigroups Representation of discrete semigroups Representations of Lie semigroups H. Hecke algebras Hecke algebras I. Invariant theory Section 7. Machine computation. Algorithms. Tables. Counting algebraic structures Some notes on this volume: Besides some general article(s) on machine computation in algebra, this volume should contain speciﬁc articles on the computational aspects of the various larger topics occurring in the main volumes, as well as the basic corresponding tables. There should also be a general survey on the various available symbolic algebra computation packages Georgy P. Egorychev, Eugene V. Zima, Integral representation and algorithms for closed form summation (5; 75 pp) The CoCoA computer algebra system Groebner bases and their applications Software for automatic groups Section 8. Applied Algebra Marlos Viana, Canonicak decompositions and invariants for data analysis (6; 16 pp) Section 9. History of algebra (see also K.T. Lam, Hamilton’s quaternions, in Section 3A) History of coalgebras and Hopf algebras Development of algebra in the 19th century

List of Contributors Alexander E. Guterman, Faculty of Algebra, Department of Mathematics and Mechanics, Moscow State University, Moscow, GSP-2, Russia, e-mail: [email protected] Kazimierz Szymiczek, Department of Mathematics, University of Silesia, Katowice, Poland, e-mail: [email protected] Ronald Brown, School of Computer Science, University of Wales, Dean St. Bangor, Gwynedd LL57 1UT, UK, e-mail: [email protected]. Katsutoshi Amano, 8-29 Ida-Sammai-cho, Kawasaki, Kanagawa 211-0037, Japan, e-mail: [email protected] Akaira Masuoka, Department of Mathematics, University of Tsukuba, Ibaraki 3058571, Japan, e-mail: [email protected] Mitsuhiro Takeuchi, Department of Mathematics, University of Tsukuba, Ibaraki 3058571, Japan, e-mail: [email protected] Gabriella Böhm, Research Institute for Particle and Nuclear Physics Budapest, POB 49, H-1525 Budapest, Hungary, e-mail: [email protected] Tomasz Brzezinski, Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK, e-mail: [email protected] Michiel Hazewinkel, Burg. s’Jacob iaan 18, 1401BR Bussum, The Netherlands, e-mail: [email protected] Jae-Hoon Kwon, Department of Mathematics, University of Seoul, 90 Cheonnong Dong, Dongdaemungu, Seoul 130-743, Korea, e-mail: [email protected] Xueqing Chen, Department of Mathematics and Computer Sciences, University of Wisconsin-Whitewater, 800 West Main Street, Whitewater, WI 53190, USA, e-mail: [email protected] Tomáš Pospíchal, Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario N9B 3P4, Canada, e-mail: [email protected] Ki-Bong Nam, Department of Mathematics and Computer Sciences, University of Wisconsin-Whitewater, 800 West Main Street, Whitewater, WI 53190, USA, e-mail: [email protected] Marlos Viana, University of Illinois at Chicago Eye Center, UIC Eye Center, 1855 West Taylor Street (m/c648), Chicago, IL 60612, USA, e-mail: viana@ gmail.com, [email protected]

xxv

Matrix Invariants over Semirings∗ Alexander E. Guterman Faculty of Algebra, Department of Mathematics and Mechanics, Moscow State University, Moscow, 119992, GSP-2, Russia E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Matrices and determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Singularity and determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Semimodules: bases and dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Relations between different rank functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Arithmetic behavior of rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Rank-sum inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Sylvester inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Ranks of matrix union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ 2000 Mathematics Subject Classiﬁcation: 15A45, 15A33, 15A03. The work is partially supported by RFBR grant N. 08-01-00693

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00201-5 3

4 5 7 10 14 15 18 22 22 26 29 31 31

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A. E. Guterman

1. Introduction In the last decades, matrices with entries from various semirings have attracted attention of many researchers working in both theoretical and applied mathematics. The concept of a semiring is deﬁned as follows: Deﬁnition 1.1. A semiring is a set S with two binary operations, addition and multiplication, such that • • • •

S is an Abelian monoid under addition (identity denoted by 0); S is a semigroup under multiplication (identity, if any, denoted by 1); multiplication is distributive over addition on both sides; s0 = 0s = 0 for all s ∈ S.

In this chapter, we always assume that there is a multiplicative identity 1 in S, which is different from 0. Brieﬂy, a semiring differs from a ring by the fact that not every element is required to have the additive inverse. The most common examples of semirings that are not rings are the non-negative integers Z+ , the non-negative rationals Q+ , and the non-negative reals R+ , with the usual addition and multiplication. There are classical examples of non-numerical semirings as well. Probably, the ﬁrst such example appeared in the work of Dedekind [1] in connection with the algebra of ideals of a commutative ring (one can add and multiply ideals, but it is not possible to subtract them). Later, Vandiver [2] put forward the concept of semirings as the best algebraic structure, which includes both rings and bounded distributive lattices. Other important examples of semirings are Boolean algebras, max-algebras, tropical semirings, and fuzzy scalars. See the monographs [3–5] for more details. It should be noted that the majority of examples of semirings arise in various applications of algebra. Among them, there are automata theory [6, 7], optimization theory [8, 9], combinatorial optimization [10–12], optimal control [13, 14], discrete event systems [15], operations research [13], ergodic control [16, 17], mathematical economics [16], assignment problem [11, 18], graph theory [11, 19], and algebraic geometry [20–22]. For a detailed index of applications, one may consult [4, pp. 355–356]. Deﬁnition 1.2. A semiring S is called commutative if the multiplication in S is commutative. Deﬁnition 1.3. A semiring S is called antinegative (or zero-sum-free) if a + b = 0 implies that a = b = 0. This means that the zero element is the only element with an additive inverse. Deﬁnition 1.4. We say that a semiring S has no zero divisors if from ab = 0 in S it follows that a = 0 or b = 0 (or both).

Matrix Invariants over Semirings

5

Deﬁnition 1.5. A semiring S is called Boolean if S is isomorphic to a suitable set of subsets of a given set M, the sum of two subsets being their union, and the product being their intersection. The zero element is the empty set and the identity element is the whole set M. It is straightforward to see that Boolean semirings are commutative and antinegative. If a Boolean semiring S consists of only two subsets of M: the empty subset and M, then it is called a binary Boolean semiring (or {0, 1}-semiring) and is denoted by B. Note that the binary Boolean semiring does not have zero divisors. Deﬁnition 1.6. A semiring is called a chain semiring if the set S is totally ordered with universal lower and upper bounds, and the operations are deﬁned by a + b = max{a, b} and a · b = min{a, b}. It is straightforward to see that any chain semiring S is a Boolean semiring on a set of appropriate subsets of S. Namely, let us consider the set M of all elements in S. We choose all those subsets of M that consist of all elements of S, which are strictly lower than a given element. Chain semirings are examples of Boolean semirings without zero divisors. Deﬁnition 1.7. A semiring is called a max-algebra if the set S is an ordered group with a multiplication ∗ and an order relation ≤, and operations in S are deﬁned as follows: a + b = max{a, b}, a · b = a ∗ b, for any a, b ∈ S. The most common example of max-algebras is Rmax := (R ∪ {−∞}, ≤, +). In a similar way, min-algebras can be deﬁned, with the only difference being a + b = min{a, b}. These algebras are sometimes called tropical algebras. 2. Matrices and determinants Let Mm,n (S) denote the set of m × n matrices with entries from the semiring S, Mn (S) = Mn,n (S). Under the natural (and usual) deﬁnitions of matrix addition and multiplication, Mn (S) is obviously a semiring. Matrix theory over semirings has been an object of intensive study during the last decades; see for example the monographs [3, 23] and references therein. For more recent results on this subject, see [20, 24–30]. The matrix In is the n × n identity matrix, Jm,n is the m × n matrix with all elements equal to 1, and Om,n is the m × n zero matrix. We omit the subscripts when the order is obvious from the context and we write I, J, and O. The matrix Ei,j m denotes the matrix with 1 at the (i, j)th position and zeros elsewhere, Ci = Ei,j is the ith column matrix, and Rj =

n i=1

j=1

Ei,j is the jth row matrix. For A ∈ Mm,n (S)

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A. E. Guterman

and B ∈ Mm,k (S), we denote by (A|B) ∈ Mm,n+k (S) the matrix whose rows are obtained by the union of columns of A and B. The development of linear algebra over semirings certainly requires an analog of the determinant function (see [4]). However, it turns out that even over commutative semirings without zero divisors, the classical determinant cannot be deﬁned as over ﬁelds and commutative rings. The main problem lies in the fact that, in semirings that are not rings not all elements possess an additive inverse. A natural replacement of the determinant function for matrices over commutative semirings is the bideterminant known since 1972 (see [31] and also [4, 32–34]). The bideterminant is useful for the solution of various pure algebraic problems (see [34]). The bideterminant is important for applications as well, for example, to solve systems of linear equations (see [32,35]), and in connections with graph theory (see [31]). For example, in [36], on the basis of the properties of max-algebraic determinants, it is shown that, the problem of the veriﬁcation if a matrix is sign-nonsingular is polynomially equivalent to the problem of deciding whether the digraph of the corresponding matrix contains a cycle of even length. See also monographs [4, 5, 35] for more detailed and self-contained information. Deﬁnition 2.1. [4, Chapter 19] The bideterminant of a matrix A = [ai,j ] ∈ Mn (S) is the pair (A+ , A− ), where A+ = a1,σ(1) · · · an,σ(n) , A− = a1,σ(1) · · · an,σ(n) . σ∈An

σ∈Sn \An

Here, Sn denotes the symmetric group on the set {1, . . . , n} and An denotes its subgroup of even permutations. It is known that the bideterminant function possesses some natural properties. Namely, it is invariant under transposition, and for any scalar α ∈ S, (αA+ , αA− ) = (αn A+ , αn A− ). However, some basic properties of the determinant are no longer true for the bideterminant. For example, if A is invertible, then A+ = A− , but the converse is not always true. Example 2.2. [32] Let us consider A = E1,1 + 2E1,2 + 3E2,1 + 4E2,2 ∈ M2 (S), where S = (Q+ , max, · ), i.e. the set of non-negative rationals with the standard multiplication and the addition deﬁned by a + b = max{a, b}. Then (A+ , A− ) = (4, 6) but A is not invertible. The bideterminant is not multiplicative in general. However, the following weaker version of this property is true. Proposition 2.3. [30] The following equality holds for all A, B ∈ Mn (S) AB+ + A+ B− + A− B+ = AB− + A+ B+ + A− B− ,

Matrix Invariants over Semirings

7

and, more generally, for A1 , . . . , As ∈ Mn (S), it holds that A1 , A2 · · · As + + A1 t1 A2 t2 · · · As ts t1 ,...,ts =± t1 ···ts =−

= A1 A2 · · · As − +

A1 t1 A2 t2 · · · As ts

t1 ,...,ts =± t1 ··· ts =+

It also appears that the following function can be used instead of the determinant for matrices over semirings. Deﬁnition 2.4. The permanent of a matrix A = [ai,j ] ∈ Mn (S) is per (A) = a1,σ(1) · · · an,σ(n) . σ∈Sn

Also, if S is a subsemiring of a certain commutative ring R, then for A ∈ Mn (S), (−1)σ a1,σ(1) · · · an,σ(n) = A+ − A− is well the determinant det(A) = deﬁned in R.

σ∈Sn

In the following, we will concentrate on the recognition of different matrix properties via the determinant function. 2.1. Singularity and determinant Let R be an associative ring. A matrix, A ∈ Mm,n (R), with entries from R is called right singular if there exists a nonzero element x ∈ Rn such that Ax = 0, and A is called left singular if there exists a nonzero element y ∈ Rm such that yt A = 0. It is proved in [37, Theorem 9.1] that both these deﬁnitions coincide for matrices over commutative rings and for square matrices are equivalent to the statement that det A is a zero divisor in R, where det A is the usual determinant of a square matrix over a commutative ring. The notion of singularity for matrices over semirings can be deﬁned in a similar way: Deﬁnition 2.5. A matrix A ∈ Mm,n (S) is S-right singular if Ax = 0 for some nonzero x ∈ S n . Deﬁnition 2.6. A matrix A ∈ Mm,n (S) is S-left singular if xt A = 0t for some nonzero x ∈ S m .

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A. E. Guterman

Deﬁnition 2.7. A matrix A ∈ Mm,n (S) is S-singular if A is either S-left singular or S-right singular. The next example shows that even over antinegative commutative semirings without zero divisors, there exist matrices that are S-left singular and are not S-right singular or vice versa. 0 0 1 0 Example 2.8. [24] Let A = ,B = ∈ M2 (Z+ ). We have that 1 1 1 0 Ax = 0 forces x = 0 since Z+ is an antinegative semiring, but [1, 0]A = [0, 0]. Similarly xt B = 0 forces x = 0 while B[0, 1]t = [0, 0]t . Deﬁnition 2.9. A matrix A ∈ Mm,n (S) is S-nonsingular if A is neither S-left singular nor S-right singular. Lemma 2.10. [24, Lemma 3.8] Let S be an antinegative semiring. Then the following conditions are equivalent for any matrix A ∈ Mm,n (S): 1. A is S-singular. 2. A has a zero row or a zero column. Note that if S is commutative and A is an S-singular square matrix, then (A+ , A− ) = (0, 0). However, the following example shows that there are S-nonsingular matrices with bideterminant equal to (0, 0). Example 2.11. [24] Over any commutative antinegative semiring, 0 0 1 1 0 0

1 0 1

+ 0 =0= 1 0

0 1 0

1 0 1

− .

However, if the semiring S is also a subsemiring of an associative ring R without zero divisors, we can consider the following notion of singularity as well. Deﬁnition 2.12. We say that a matrix A ∈ Mm,n (S) is R-right singular if Ax = 0 for some nonzero x ∈ Rn . Deﬁnition 2.13. We say that a matrix A ∈ Mm,n (S) is R-left singular if xt A = 0 for some nonzero x ∈ Rm . It is straightforward to see that if a semiring S is a subsemiring of a certain ring R, then R-right (left) singularity follows from S-right (left) singularity. However, the following example shows that there are S-nonsingular matrices, which are R-singular.

Matrix Invariants over Semirings

Example 2.14. For any n, the matrix Jn = and Z-right singular but Z+ -nonsingular.

n i,j=1

9

Ei,j ∈ Mn (Z+ ) is Z-left singular

Deﬁnition 2.15. We say that a matrix A ∈ Mm,n (S) is R-singular if A is R-left singular or R-right singular. Deﬁnition 2.16. A matrix A ∈ Mm,n (S) is R-nonsingular if A is not R-singular. Note that similarly to the situation over ﬁelds, all nonsquare matrices are R-singular. Assume in addition that R is commutative. Then det(A) ∈ R is well deﬁned and the following analog of [37, Theorem 9.1] can be obtained. Lemma 2.17. [24, Lemma 5.3] If R is an associative commutative ring then for a matrix A ∈ Mn (S) the following conditions are equivalent: 1. A is R-right singular; 2. A is R-left singular; 3. det A = 0. Deﬁnition 2.18. An element g from a certain multiplicative system G with identity element 1G is invertible (resp., left invertible, right invertible) if there is an element f ∈ G such that fg = gf = 1G (resp., fg = 1G , gf = 1G ). Deﬁnition 2.19. A matrix A ∈ Mn (S) is called monomial if it has exactly one nonzero element in each row and column. It follows from [38] that a matrix over an antinegative semiring is invertible (resp., left invertible, right invertible) if and only if it is a monomial matrix such that all its nonzero elements are invertible (resp., left invertible, right invertible). For matrices over max-algebras, the following notion of singularity is often in use (see [28] for the details). Deﬁnition 2.20. Let S be a max-algebra (operations are denoted by max and +). A matrix A = [aij ] ∈ Mn (S) is said to be tropically singular if the maximum in the expression for the permanent per (A) = max{a1σ(1) + · · · + anσ(n) } σ∈Sn

is achieved at least twice. This can be generalized to the case of an arbitrary antinegative semiring S in the following way.

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A. E. Guterman

Deﬁnition 2.21. A matrix A = [aij ] ∈ Mn (S) is said to be tropically singular if there exists a subset T ∈ Sn such that a1σ(1) · · · anσ(n) = a1σ(1) · · · anσ(n) . σ∈T

σ∈Sn \T

We use these concepts in Section 4.1 to deal with the notion of rank. 3. Semimodules: bases and dimension Semimodules over semiring are analogs of vector spaces over ﬁelds and modules over rings. These are discussed more precisely in the following. Deﬁnition 3.1. Given a semiring S, we deﬁne a left semimodule, U, over S to be an abelian monoid with identity element, 0 ∈ U. In addition, U is equipped with a function, S ×U (s, u)

→ →

U su,

called scalar multiplication such that for all u and v in U and r, s ∈ S we have 1. 2. 3. 4. 5.

(sr)u = s(ru), (s + r)u = su + ru, s(u + v) = su + sv, 1u = u, s0 = 0 = 0u.

In a similar way, a right semimodule can be deﬁned. In particular, under the natural deﬁnitions of matrix addition and multiplication with scalars, Mm,n (S) is obviously a semimodule over S. Deﬁnition 3.2. An element u in a semimodule U is called a left (resp., right) linear combination of elements from a certain subset P ⊆ U if there exists k ∈ N, k k s1 , . . . , sk ∈ S, u1 , . . . , uk ∈ P such that u = si ui resp., u = ui si . In this i=1 i=1 k k case, si ui ui si is called a left (right) linear combination of the elements i=1

i=1

u1 , . . . , uk from P with coefﬁcients s1 , . . . , sk from S. Deﬁnition 3.3. A left (right) linear span, P S , of the set P is the set of all left (right) linear combinations of elements from P with coefﬁcients from S. We say that the set P generates a subset V ⊆ U if V ⊆ P S . Note that by deﬁnition, all linear combinations, we consider, are ﬁnite and nonempty. We will not specially point out the semiring of the coefﬁcients if it will be clear from the context.

Matrix Invariants over Semirings

11

As over ﬁelds and rings, we denote the linear span of the set {[1, 0, . . . , 0]t , [0, 1, 0, . . . , 0]t , . . . , [0, . . . , 0, 1]t } by S n ; here, At denotes the transposed matrix to the matrix A. Note that, in contrast with vector spaces over ﬁelds, there are several ways to deﬁne a notion of independence for semimodules. For example, in [27,37,39], the following deﬁnition is used. Deﬁnition 3.4. A set of elements, P, from a semimodule U over a semiring S is called left (right) linearly independent if there is no element in P that can be expressed as a left (right) linear combination of the other elements of P with the coefﬁcients from S. The set P is called linearly dependent if it is not linearly independent. However, in [16, 19, 32], the following deﬁnition is used. Deﬁnition 3.5. A system of elements, u1 , . . . , uk , in a semimodule U is left (resp., right) linearly dependent in the Gondran–Minoux sense if there exist two subsets I, J ⊆ K := {1, . . . , k}, I ∩ J = and scalars ∅, I ∪ J = K, α1 , . . . , αk ∈ S, not all equal to 0, such that αi ui = αj uj resp., ui αi = u j αj . i∈I

j∈J

i∈I

j∈J

Remark 3.6. For any semiring, this deﬁnition is stronger than the ﬁrst one. Namely, any system of elements, which is independent in the sense of Deﬁnition 3.5, is evidently independent in the sense of Deﬁnition 3.4, but the converse is not true as the following example shows. Example 3.7. It is shown in [40, Lemma 3.11] that the system ⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 1 0 1 0 ⎬ ⎨ u1 = ⎝ 0 ⎠ , u2 = ⎝ 1 ⎠ , u3 = ⎝ 0 ⎠ , u4 = ⎝ 1 ⎠ ⎩ ⎭ 0 0 1 1 is linearly independent in the sense of Deﬁnition 3.4 over any antinegative semiring. However, u1 + u4 = u2 + u3 , that is, this system is linearly dependent in the sense of Deﬁnition 3.5. In [41], the following deﬁnition of linear dependence is introduced: Deﬁnition 3.8. A system of elements, u1 , . . . , uk , ui = [u1i , . . . , uni ]t , i = 1, . . . , k, in a semimodule U is left (resp., right) linearly dependent in the sense of Izhakian if ∪ Jl = K, there exist two series of subsets Il , Jl ⊆ K := {1, . . . , k}, Il ∩ Jl = ∅, Il αi uli = l = 1, . . . , n, and scalars α1 , . . . , αk ∈ R, not all equal to 0, such that i∈Il l l l αj uj resp., ui αi = uj αj for all l, 1 ≤ l ≤ n. A system is called

j∈Jl

i∈Il

j∈Jl

linearly independent if it is not linearly dependent.

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Remark 3.9. Deﬁnition 3.8 is stronger than Deﬁnition 3.5. Namely, it is straightforward to see that a system of elements, which is independent in the sense of Deﬁnition 3.8, is also independent in the sense of Deﬁnition 3.5; however, the converse is not always true as it is shown by the following example. Example 3.10. Let S be an arbitrary antinegative semiring. Let us consider the system of vectors [0, 1, 1]t , [1, 0, 1]t , [1, 1, 0]t with the entries in S. By the antinegativity, it is easy to see that these vectors are linearly independent in the sense of Deﬁnitions 3.4 and 3.5. However, taking I1 = {1, 2}, J1 = {3}, I2 = {1, 2}, J2 = {3}, I3 = {1}, J3 = {2, 3}, and the coefﬁcients (1, 1, 1), we see that these vectors are linearly dependent in the sense of Deﬁnition 3.8. As in the case of vector spaces over ﬁelds, we can introduce the following deﬁnition. Deﬁnition 3.11. A collection, B, of elements is called a left (right) basis of the semimodule M if these elements are left (right) linearly independent in the sense of Deﬁnition 3.4 and their left (right) linear span is M. Remark 3.12. It may seem that we can have three different deﬁnitions of basis if we use here Deﬁnition 3.5 or Deﬁnition 3.8 instead of Deﬁnition 3.4. However, it appears that this is not the case. Namely, in the following, we show that any ﬁnitely generated module has a basis in the sense of Deﬁnition 3.11, but the same fact does not hold if we consider Deﬁnition 3.5 or Deﬁnition 3.8 instead of Deﬁnition 3.4 (see Remark 3.14). Moreover, it is easy to see that if we consider similar deﬁnitions of basis using the notion of independence from Deﬁnition 3.5 or 3.8, then any B, which is a basis in this sense, is a basis in the sense of Deﬁnition 3.11. In the following we will consider left linear independence and left basis without pointing this out anymore. However, all those considerations are valid for right linear independence and right basis as well. Lemma 3.13. Let S be a semiring and U be a ﬁnitely generated semimodule over S. Then for the notion of linear dependence introduced in Deﬁnition 3.4, there exists a basis of U. Proof. Let M be a certain ﬁnite generating set for U. If M is linearly independent in the sense of Deﬁnition 3.4, then we are done. Assume that M is linearly dependent. Then there exist m ∈ M, m1 , . . . , mk ∈ M, mi = m for i = 1, . . . , k, and s1 , . . . , k si mi . It follows that M1 = M \ {m} is a generating set sk ∈ S such that m = i=1

for U also. Continuing this process, we complete the proof.

Matrix Invariants over Semirings

13

Remark 3.14. Let us show that there are spaces without a basis if we consider linear independence in the sense of Deﬁnition 3.5 or Deﬁnition 3.8. We consider the linear span U4 of the system of vectors ⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ 1 0 1 0 ⎬ ⎨ u1 = ⎝ 0 ⎠, u2 = ⎝ 1 ⎠, u3 = ⎝ 0 ⎠, u4 = ⎝ 1 ⎠ ⎩ ⎭ 0 0 1 1 (the same vectors as in Example 3.7). Assume that there is a basis in this space in the sense of Deﬁnition 3.5. Thus, it is a basis in the sense of Deﬁnition 3.4 by Remark 3.6. However, it is proved in [40, Lemma 3.11] that any basis of U4 contains u1 , u2 , u3 , and u4 , which are linearly dependent in the sense of Deﬁnition 3.5. The same considerations hold for linear dependence in the sense of Deﬁnition 3.8. In [27], it is proved that if S is a max-algebra, U is a ﬁnitely generated semimodule over S, then any basis of U in the sense of Deﬁnition 3.4 has the same number of elements. The following example shows that this does not hold for more general classes of semirings, even under the condition of antinegativity. √ √ Example 3.15. Let S = Z[ 7]+ be the set of non-negative reals of the form a+b 7, where a, b ∈ Z. √ We√consider the semimodule √ U√over S, which is generated by the elements {3 − 7, 7 − 2}. The set {3 − 7, 7 − 2} is a basis √ for

U √in the sense of Deﬁnition 3.4, which consists of two elements, but 1 = 3 − 7 + 7 − 2 also generates U over S. Thus, {1} is a basis consisting of one element. In view of these examples, we give the following deﬁnition. Deﬁnition 3.16. Let U be a semimodule over a certain semiring S. The (left) dimension, dim(U), is the minimal number of elements in any (left) basis of U. By Lemma 3.13, any ﬁnitely generated semimodule has ﬁnite dimension. However, the following example shows that even a three-dimensional semimodule may have an inﬁnite-dimensional subsemimodule. Example ⎡ ⎤ 3.17. It is shown in [27] that in the sense of Deﬁnition 3.4, the vectors xi ⎣ 0 ⎦ ∈ R3max , i = 1, 2, . . . , m, are linearly independent for any m for differ−xi ent xi . Thus, in some cases, it is more informative to consider the maximal number of linear independent vectors in the system than to consider its dimension. From this point of view, all three deﬁnitions, Deﬁnitions 3.4, 3.5, and 3.8, work. It is easy to see that if S is a subsemiring of a certain ﬁeld, then any n + 1 vectors in S n are linearly

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dependent in the sense of Deﬁnition 3.5. Also, it is proved in [42] that any n + 1 vectors in Rnmax are linearly dependent in the sense of Deﬁnition 3.5. Hence, in both cases, they are dependent in the sense of Deﬁnition 3.8. In particular, a situation as in Example 3.17 cannot appear in these cases. 4. Rank functions Deﬁnition 4.1. We say that a matrix, A, dominates a matrix, B, A ≥ B, if and only if bi,j = 0 implies that ai,j = 0. Deﬁnition 4.2. If A = [ai,j ], B = [bi,j ] ∈ Mm,n (S), A ≥ B, we denote by A\B the matrix C = [ci,j ] ∈ Mm,n (S) such that 0 if bi,j = 0 ci,j = . ai,j otherwise Many authors have investigated various rank functions for matrices over semirings and their properties [40, 44–53]. It is known that over a ﬁeld, the concept of a matrix rank has a geometrical interpretation as the dimension of the image space of the corresponding linear transformation. Over semirings, the situation is more involved, namely, this geometric approach leads to some surprising properties of the corresponding rank function. For example, it can appear that the rank of a submatrix is greater than the rank of a matrix. The reason is that the dimension of a certain subsemimodule may exceed in general the dimension of the whole semimodule, cf. Example 3.17 for matrices with entries from a max-algebra and Example 3.7. For completeness, we give it here again in more details: Example 4.3. [40] Let S be an arbitrary antinegative semiring without zero divisors. Consider the semimodule U4 over S generated by the vectors a1 = [0, 1, 0]t , a2 = [0, 0, 1]t , a3 = [1, 0, 1]t , and a4 = [1, 1, 0]t from Example 3.7. Then dim U4 = 4, however, U4 is a proper subsemimodule of the three-dimensional S-semimodule S 3 . Indeed, it is easy to see that none of these vectors is a linear combination of the others. The reason why U4 has no basis of less than four elements is that [1, 0, 0]t cannot be expressed as a linear combination of vectors from U4 due to the antinegativity (see also [40, Lemma 3.11] for the detailed proof). Based on Example 4.3, we may easily construct the following matrix example: Example 4.4. Let S be any antinegative semiring without zero divisors; we consider the matrix ⎡ ⎤ 1 0 0 1 1 Y = ⎣0 1 0 0 1⎦ 0 0 1 1 0

Matrix Invariants over Semirings

and its proper submatrix ⎡ 0 0 1 X = ⎣1 0 0 0 1 1

15

⎤ 1 1 ⎦. 0

The image of the linear operator corresponding to Y is the whole space S 3 ; so its dimension is three, however, the dimension of the image of the linear operator corresponding to X is equal to 4 by Example 4.3. In the following, we collect different deﬁnitions of a matrix rank over semirings. Each of them has some advantages and disadvantages in comparison with the usual rank function over a ﬁeld. However, all these functions are vital to solve some problems arising in the theory of matrices over semirings and their applications. 4.1. Deﬁnitions We start with one of the most important semiring rank functions. Deﬁnition 4.5. A matrix A ∈ Mm,n (S) is of factor rank k (f (A) = k) if there exist matrices B ∈ Mm,k (S) and C ∈ Mk,n (S) with A = BC, and k is the smallest positive integer for which such a factorization exists. By deﬁnition, the only matrix with the factor rank zero is the zero matrix, O. Let us note (see [45, 54] for the proof ) that the factor rank of A is equal to the minimum number of factor rank-1 matrices whose sum is A. For any submatrix B of A, it holds that f (B) ≤ f (A) [49]. The notion of factor rank is very important in different applications. It was used in [55] for demographical investigations, in [56] for combinatorial optimization problems, and in [54] for statistics. If S is a subsemiring of a certain ﬁeld, then there is the usual rank function ρ(A) for any matrix A ∈ Mm,n (S). The geometric approach leads to the following deﬁnitions based on the notion of linear independence. Deﬁnition 4.6. The matrix A ∈ Mm,n (S) is of row rank k (r(A) = k) if the dimension of the linear span of the rows of A is equal to k. Deﬁnition 4.7. The matrix A ∈ Mm,n (S) is of column rank k (c(A) = k) if the dimension of the linear span of the columns of A is equal to k. It is proved in [49] that r(B) ≤ r(A) if B is obtained by deleting some columns of A and c(B) ≤ c(A) if B is obtained by deleting some rows of A. However, as

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Example 4.4 shows, in general, the rank of a submatrix can be greater than the rank of a matrix. The following example shows that all aforesaid functions may differ for matrices over semirings. Example 4.8. [39] Let ⎡ 2 3 0 ⎢0 0 2 ⎢ A=⎢ ⎢0 0 2 ⎣0 0 2 2 3 1

0 2 4 0 0

0 2 0 4 2

⎤ 1 2⎥ ⎥ 2⎥ ⎥ ∈ M5,6 (Z+ ). 2⎦ 2

Then ρ(A) = 3, f (A) = 4, r(A) = 5, c(A) = 6. Since a basis of the linear span of a system of elements may not lie in this system (Examples 5.3 and 5.4 below), we consider also the following deﬁnitions of row and column ranks. Deﬁnition 4.9. The matrix A ∈ Mm,n (S) is said to be of spanning row rank k (sr(A) = k) if the minimal number of rows that span all rows of A is k. Deﬁnition 4.10. The matrix A ∈ Mm,n (S) is said to be of spanning column rank k (sc(A) = k) if the minimal number of columns that span all columns of A is k. The following deﬁnitions depend on Deﬁnitions 3.4, 3.5, or 3.8 of linear independence, correspondingly. Deﬁnition 4.11. The matrix A ∈ Mm,n (S) is said to be of maximal row rank k (mr 1 (A) = k, mr 2 (A) = k, or mr 3 (A) = k) if it has k linearly independent rows, in the sense of Deﬁnitions 3.4, 3.5, or 3.8, respectively, and any (k + 1) rows are linearly dependent. Deﬁnition 4.12. The matrix A ∈ Mm,n (S) is said to be of maximal column rank k (mci (A) = k, i = 1, 2, 3) if it has k linearly independent columns and any (k + 1) columns are linearly dependent with respect to Deﬁnitions 3.4, 3.5, or 3.8. Also to avoid the obstruction of Example 4.4, we may use the following deﬁnitions: Deﬁnition 4.13. The matrix A ∈ Mm,n (S) is said to be of enveloping row rank k (er(A) = k) if the minimal dimension of any semimodule containing all rows of A is equal to k. Deﬁnition 4.14. The matrix A ∈ Mm,n (S) is said to be of enveloping column rank k (ec(A) = k) if the minimal dimension of any semimodule containing all columns of A is equal to k.

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17

It will be shown later (see Proposition 5.1) that actually er(A) = ec(A) = f (A) for matrices over arbitrary semirings; r(A) = mr 1 (A) = sr(A) [resp., c(A) = mc1 (A) = sc(A)] if S is a max-algebra [27]. Let us recall that for matrices over ﬁelds, all notions introduced earlier give the same concept; however, they can be different for matrices over semirings. In the following sections, we provide some examples illustrating these difference and investigate how these functions are related to each other. The deﬁnition of rank over ﬁelds based on the notion of maximal nonsingular minors has the following analogs in the semiring case: Deﬁnition 4.15. Let A ∈ Mm,n (S). It is said that A is of tropical rank k (trop (A) = k) if k is the largest number such that A has a (k × k)-submatrix, which is tropically nonsingular (see Deﬁnition 2.21). Deﬁnition 4.16. Let A ∈ Mm,n (S). It is said that A is of symmetric rank or symmetric tropical rank k (rk sym (A) = k) if k is the largest number such that A has a (k × k)-submatrix A , satisfying |A |+ = |A |− . Remark 4.17. In other words, symmetric tropical rank differs from the tropical rank by ﬁxing T from Deﬁnition 2.21 to be the alternating group Ai ⊂ Si for all (i × i)submatrices of A, i = 1, . . . , n. Proposition 4.18. Let A ∈ Mm,n (S). Then sym(A) ≤ trop(A). Proof. The proof follows easily from Remark 4.17.

Now we consider the so-called combinatorial ranks, which are very useful for graph theory, transversal theory, and communication networks (see [4, 57, 58] and references therein). Note that these ranks do not coincide with the usual rank function even in the case where S is a ﬁeld. Deﬁnition 4.19. A line of a matrix is its row or column. Deﬁnition 4.20. A matrix A ∈ Mm,n (S) is of term rank k (t(A) = k) if the minimum number of lines needed to include all nonzero elements of A is equal to k. Let us denote by tc (A) the least number of columns needed to include all nonzero elements of A and by tr (A) the least number of rows needed to include all nonzero elements of A. Deﬁnition 4.21. A generalized diagonal of a matrix A ∈ Mm,n (S) is a set of min{m, n} positions in A such that no row or column contains two of these positions.

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Proposition 4.22 (König Theorem). [57, Theorem 1.2.1] Let S be a semiring, A ∈ Mm,n (S). Then the term rank of A is the maximum number of nonzero entries in some generalized diagonal of A. Also the term rank can be characterized in terms of a zero submatrix of maximal size. Proposition 4.23. [59] Let S be a semiring, A ∈ Mm,n (S). Then t(A) = k if and only if there exist positive integers s, r, s + r = n + m − k such that for a certain permutation of rows and columns of A contains a submatrix Or,s and no permutation of rows and columns of A contains an Op,q if p + q > n + m − k. The following “dual” matrix invariant is also known. Deﬁnition 4.24. The matrix A ∈ Mm,n (S) is said to be of zero-term rank k (z(A) = k) if the minimum number of lines needed to include all zero elements of A is equal to k. In the case when S is the max-algebra, Rmax , also the notion of Kapranov rank, K(A), is useful (see [28, 60] for the details). We should note that the list of rank functions introduced here is not exhausting. We presented here only the most common deﬁnitions. However, there are still other natural ways to deﬁne rank functions over semirings. 5. Relations between different rank functions If the semiring S coincides with a ﬁeld, then ρ(A) = f (A) = r(A) = c(A) = sr(A) = sc(A) = mr i (A) = mci (A), i = 1, 2. Over more general semirings, the situation is more complicated. Proposition 5.1. Let A ∈ Mm,n (S), then er(A) = f (A) = ec(A). Proof. Let us check that er(A) ≤ f (A). Write f (A) = f . Then A = BC for certain B ∈ Mm,f (S) and C ∈ Mf,n (S). Then the row space of A is contained in the row space of C; hence er(A) ≤ er(C) ≤ f . Let us show that the converse inequality also holds. We write er(A) = r. By deﬁnition, there exist vectors v1 , . . . , vr ∈ S n and elements αi,j ∈ S, i = 1, . . . , m, r αi,j vjt . Thus A = BC, where j = 1, . . . , r such that the ith row of A is equal to j=1 ⎤ ⎡ v1 ⎥ ⎢ B = [αi,j ], C = ⎣ ... ⎦. vr Similar considerations with the columns of A show that ec(A) = f (A).

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Proposition 5.2. Let A ∈ Mm,n (S). Then r(A) ≤ sr(A) ≤ mr 1 (A) ≤ m and ec(A) ≤ c(A) ≤ sc(A) ≤ mc1 (A) ≤ n. Proof. These statements follow directly from the deﬁnitions of these rank functions. The maximal column rank associated with Deﬁnition 3.4 may actually exceed the spanning column rank, and the spanning column rank may exceed the column rank over some semirings shown in the following examples. √ √ √ Example 5.3. [51] Let √ us consider √ A = [3 − 7, 7 − √2] ∈ M √1,2 (Z[ 7]+ ).√Thus sc(A) = 2, since 3 − 7 = α( 7 − √ 2) and√α(3 − 7) = 7 − 2 in Z[ 7]+ . However, c(A) = 1 since 1 = (3 − 7) + ( 7 − 2) generates the column space of A. √ √ √ Example 5.4. [45] We consider A = [4 − 7, 7 − 2, 1] ∈ M1,3 (Z[ 7]+ ). Thus sc(A) = 1, since 1 spans all columns of A. However, just like in the previous example, one can see that mc(A) = 2. The same statements are also true for the row ranks. Proposition 5.5. Let A ∈ Mm,n (S) then f (A) ≤ min{r(A), c(A)}. Proof. See [45, Proposition 3.1.1].

Let us note that this inequality can be strict. Moreover, unlike the case with rank over ﬁelds, row and column ranks may not coincide even over commutative semirings; moreover, column ranks may be greater than the number of rows, just like row ranks may be greater than the number of columns. Example 5.6. Let S be any antinegative semiring without zero divisors; we consider the matrix X deﬁned in Example 4.4. Then it follows from Example 4.3 and Proposition 5.2 that r(X) = sr(X) = mr 1 (X) = 3 = m while c(X) = sc(X) = mc1 (X) = 4 = n > m. Proposition 5.7. Let A ∈ Mm,n (S) then f (A) ≤ t(A). Proof. See [45, Proposition 3.1.3].

Note that the inequality max{r(A), c(A)} ≤ t(A) does not hold in general. Example 5.8. For the matrix X given in Example 4.4, one has that r(A) = t(A) = 3, c(A) = 4.

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The inequality min{r(A), c(A)} ≤ t(A) does not hold over Z+ . Example 5.9. [45] For ⎡ ⎤ 3 5 7 A = ⎣ 5 0 0 ⎦ ∈ M3,3 (Z+ ), 7 0 0 one has that r(A) = c(A) = 3, t(A) = 2. Proposition 5.10. Let S be either a max-algebra or an antinegative subsemiring of a ﬁeld, or some other semiring, for which any n + 1 vectors from S n are linearly dependent, A ∈ Mm,n (S). Then mr 2 (A) ≤ f(A). Proof. Repeat the arguments from [42, Lemma 5.9].

The factor rank of some matrices can be greater than the Gondran–Minoux rank. Example 5.11. Let us consider the matrix ⎡ ⎤ 0 1 1 1 ⎢1 0 1 1⎥ ⎥ K4 := ⎢ ⎣ 1 1 0 1 ⎦ ∈ M4 (Rmax ) or K4 ∈ M4 (B). 1 1 1 0 Then it is easy to see that the sum of ﬁrst two rows (columns) is the vector of all ones, and it is equal to the sum of last two rows (columns). Thus mr 1 (K4 ) ≤ 3. However, f (K4 ) = 4 (see [42] or [28]). Proposition 5.12. Let S be a max-algebra, A ∈ Mm,n (S). Then rk sym (A) ≤ mr 2 (A). Proof. The result follows from [19]; see also [45, Theorem 1’].

Proposition 5.13. Let S be an antinegative subsemiring of a ﬁeld, A ∈ Mm,n (S). Then mr 2 (A) = ρ(A) = rk sym (A). Proof. Let us prove the ﬁrst equality. Let some vectors over a semiring S ⊂ F be linearly dependent in the sense of Deﬁnition 3.5. Here, F is a ﬁeld. Then it is straightforward to see that they are linearly dependent over F, i.e., mr 2 (A) ≤ ρ(A) for any A. Conversely, if v1 , . . . , vk are F-linearly dependent, then they are linearly dependent in the ﬁeld of fractions of the ring S ∪ (−S), since over ﬁelds dimension does not depend on the ﬁeld extension. Thus, multiplying by the common denominator of all

Matrix Invariants over Semirings

21

coefﬁcients, we get a linear relation between v1 , . . . , vk over S ∪(−S). It is nontrivial, since char(F) = 0 by the antinegativity of S. Now, taking the summands with negative coefﬁcients to the other side of the equality sign, we obtain that v1 , . . . , vk are linearly dependent in the sense of Deﬁnition 3.5, i.e., ρ(A) ≤ mr 2 (A), which completes the proof of the ﬁrst inequality. ˆ of A, i.e., this is a maximal Now, ρ(A) is the size of a maximal nonzero minor |A| ˆ − , which proves the second inequality. ˆ + = |A| submatrix with |A| The following is the main result of [41], see also [42] and [43]. Proposition 5.14. Let S be a max-algebra, A ∈ Mm,n (S). Then mr 3 (A) = trop(A) = mc3 (A). Proposition 5.15. Let A ∈ Mm,n (S), then mr 3 (A) ≤ mr 2 (A) ≤ mr 1 (A). Proof. This follows from Remarks 3.6 and 3.9.

Now, let us show that mr 3 (A) can be less than mr 2 (A) both over max-algebras and antinegative subsemirings of ﬁelds. Example 5.17. In the case where S is a max-algebra, let us consider the matrix ⎡ ⎤ 0 1 1 K3 = ⎣ 1 0 1 ⎦ . 1 1 0 Then the rows (columns) of this matrix are linearly dependent in the sense of Deﬁnition 3.8 with the coefﬁcients 0, 0, 0. Let us recall that 0 is the identity element of a maxalgebra; so this is nontrivial linear dependence. However, |K3 |+ = 3 = 2 = |K3 |− and by Proposition 5.12, we have that mr 2 (K3 ) ≥ rk sym (K3 ) = 3. In the case where S is the semiring of nonnegative integers with natural addition and multiplication, we consider the same matrix K3 . Then its rows (columns) are linearly dependent in the sense of Deﬁnition 3.8 with the coefﬁcients 1, 1, 1; however, det(K3 ) = 2 evaluated over Z; thus, by Proposition 5.13, it holds that mr 2 (A) = 3. The statement below follows easily from Remark 3.12 and the above inequalities. Corollary 5.17. • If S is a max-algebra and the row space of A has a basis for linear independence as deﬁned by Deﬁnition 3.8, then mr 3 (A) = trop(A) = rk sym (A) = K(A) = mr 2 (A) = mc2 (A) = f (A) = r(A) = c(A) = mr 1 (A) = mc1 (A) = sr(A) = sc(A).

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• If S possesses the property that any n + 1 vectors from S n are linearly dependent and that the row space of A has a basis for linear independence as deﬁned by Deﬁnition 3.5, then mr 2 (A) = f (A) = r(A). In [47], the authors investigate how the column rank and the factor rank of matrices over certain algebraic systems change as the algebraic system changes. For different classes of semirings in [49], upper bounds are found for the sets of numbers r such that for all A ∈ Mm,n (S) with c(A) = r, it follows that f (A) = r. It appears that these upper bounds depend considerably on the semiring S. Remark 5.18. It is straightforward to see that if S = B is a binary Boolean semiring, then z(A) = t(J \ A) for any A ∈ Mm,n (S). 6. Arithmetic behavior of rank The behavior of the usual rank function ρ over ﬁelds with respect to matrix multiplication, addition, and union is given by the following classical inequalities: The rank-sum inequalities: | ρ(A) − ρ(B) |≤ ρ(A + B) ≤ ρ(A) + ρ(B); Sylvester’s laws: ρ(A) + ρ(B) − n ≤ ρ(AB) ≤ min{ρ(A), ρ(B)}; the rank inequalities for matrix union: max{ρ(A), ρ(B)} ≤ ρ(A|B) ≤ ρ(A) + ρ(B), where A and B are conformal matrices with entries from a ﬁeld. Following [45], we investigate the behavior of different rank functions over semirings with respect to matrix addition and multiplication. It turns out that the arithmetic properties of semiring rank functions depend deeply on the structure of the semiring of entries. Deﬁnition 6.1. We say that a bound is exact if there are matrices such that equality holds and best possible if for any given r and s, there are matrices of ranks r and s, respectively, such that the equality holds. 6.1. Rank-sum inequalities Let us show that the standard lower bound for the rank of a sum of two matrices is not valid in general.

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23

Example 6.2. [45] Let S be a Boolean semiring. Then it is possible that f (A + B) ≥ |f (A) − f (B)|. ⎡ 0 1 1 ⎢1 0 1 ⎢ ⎢1 1 0 ⎢ Let us consider A = K7 = ⎢ ⎢1 1 1 ⎢1 1 1 ⎢ ⎣1 1 1 1 1 1

1 1 1 0 1 1 1

1 1 1 1 0 1 1

1 1 1 1 1 0 1

⎤ 1 1⎥ ⎥ 1⎥ ⎥ 1⎥ ⎥ and B = I7 , then 1 = f (J7 ) = 1⎥ ⎥ 1⎦ 0

f (A + B), but over any Boolean semiring, f (K7 ) ≤ 5 due to the factorization ⎡ ⎤ 1 0 1 0 1 ⎡ ⎤ ⎢1 1 0 0 0⎥ 0 0 1 1 1 1 0 ⎢ ⎥ ⎢0 1 0 1 1⎥⎢1 0 0 0 0 1 1⎥ ⎢ ⎥⎢ ⎥ ⎥⎢ ⎥ K7 = ⎢ ⎢0 1 1 1 0⎥⎢0 1 1 0 0 0 0⎥ ⎢0 1 1 0 1⎥⎣1 1 0 0 1 0 0⎦ ⎢ ⎥ ⎣0 0 1 1 1⎦ 0 0 0 1 0 0 1 1 0 1 1 0 (see [61] for more details). Thus, |f (A) − f (B)| = f (B) − f (A) ≥ 7 − 5 = 2 > 1 = f (A + B). However, the following bounds are true. Proposition 6.3. [45] Let S be an antinegative semiring, A, B ∈ Mm,n (S). Then, 1. f (A + B) ≤ min{f (A) + f (B), m, n}; ⎧ ⎨ f (A) if B = O f (B) if A = O 2. f (A + B) ≥ . ⎩ 1 if A = O and B = O These bounds are exact, the upper bound is best possible and the lower bound is best possible over Boolean semirings. In the case when S is a subsemiring of R+ , the positive reals with the usual operations, the standard lower bound for the rank of sum of two matrices is again not valid because of the following example. Example 6.4. [45] Let r, s ≥ 4 and s < n − 4. Let us consider ⎡ ⎤ 1 2 3 4 ⎢1 1 1 1⎥ ⎥ A = ⎢ ⎣1 0 1 0⎦ 0 0 2 2

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and

⎡

0 ⎢ 0 B = ⎢ ⎣0 0

0 0 1 1

0 0 0 0

⎤ 0 0⎥ ⎥. 1⎦ 1

Note that f (A ) = 4, ρ(A ) = 3, f (B ) = ρ(B ) = 1, and f (A + B ) = ρ(A + B ) = 2. Let ⎤ ⎡ A O4,r−4 O4,n−r Lr−4 Or−4,n−r ⎦ A = ⎣ Or−4,4 Om−r,4 Om−r,r−4 Om−r,n−r and

⎡

⎤ O4,1 O4,s−1 O4,n−s−4 Os−1,1 Us−1 Os−1,n−s−4 ⎦, B = ⎣ Os−1,4 Om−s−3,4 Om−s−3,1 Om−s−3,s−1 Om−s−3,n−s−4 Ei,j , Lk = Ei,j ∈ Mk (S). Here, we have that f (A) = r, where Uk = B

1≤i≤j≤k

1≤j
ρ(A) = r − 1, f (B) = ρ(B) = s, and f (A + B) =| r − s | −1 < | f (A) − f (B) | if S is a subsemiring of R+ . Note that if r = s + 3, reversing the roles of A and B in A and B also gives a corresponding example. However, the following is true. Proposition 6.5. [45] Let S ⊆ R+ , A, B ∈ Mm,n (S). Then f (A + B) ≥ |ρ(A) − ρ(B)|. This bound is exact and best possible. The following inequalities are true for the term rank. Proposition 6.6. [45] Let S be an arbitrary semiring. For any matrices A, B ∈ Mm,n (S), we have t(A + B) ≤ min{t(A) + t(B), m, n}. This bound is exact and best possible. The following example shows that a nontrivial, additive lower bound does not hold over an arbitrary semiring. Example 6.7. [45] Let A = B = Jm,n over a ﬁeld whose characteristic is equal to 2. Then t(A + B) = t(O) = 0.

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25

However for antinegative semirings there is a lower bound, which is even better than the standard one. Namely, the following is true. Proposition 6.8. [45] Let S be an antinegative semiring. For any matrices A, B ∈ Mm,n (S), the following inequality holds: t(A + B) ≥ max{t(A), t(B)}. This bound is exact and best possible. Proposition 6.9. [45] Let S be an antinegative semiring. For A, B ∈ Mm,n (S), one has that 0 ≤ z(A + B) ≤ min{z(A), z(B)}. These bounds are exact and best possible. Proof. Both bounds follow directly from the deﬁnition of the zero-term rank function. To check that the lower bound is exact and best possible for eachpair (r, s), r 0 ≤ r, s ≤ min{m, n}, let us consider the family of matrices Ar = J \ Ei,i , i=1 s s−1 Bs = J \ Ei,i+1 if s < min{m, n}, and Bs = J \ i=1 Ei,i+1 + Es,1 if i=1

s = min{m, n}. Then z(Ar ) = r, z(Bs ) = s by the deﬁnition and z(Ar + Bs ) = 0 by antinegativity. Similarly, it can be checked that the upper bound is exact and best possible. As the following example shows, the standard upper bound for the additive inequalities is not valid for row and column ranks. Example 6.10. Consider ⎤ ⎡ 1 0 0 ⎢0 1 0⎥ ⎥ ⎢ ⎥ A=⎢ ⎢1 1 0⎥, ⎣1 0 0⎦ 0 1 0

⎡

0 ⎢0 ⎢ B=⎢ ⎢0 ⎣0 0

2 2 0 6 4

⎤ 2 0⎥ ⎥ 2⎥ ⎥. 2⎦ 6

Then it is easy to see that r(A) = r(B) = sr(A) = sr(B) = mr 1 (A) = 2. However, ⎡ ⎤ 1 2 2 ⎢0 3 0⎥ ⎢ ⎥ ⎥ r(A + B) = r ⎢ ⎢ 1 1 2 ⎥ = 5 = sr(A + B) = mr 1 (A + B) ⎣1 6 2⎦ 0 5 6 over Z+ .

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Proposition 6.11. [45] Let S be an antinegative semiring. Then for O = A, B ∈ Mm,n (S), one has that 1 ≤ c(A + B), r(A + B), sr(A + B), sc(A + B), mr i (A + B), mci (A + B), i = 1, 2, 3. These bounds are exact over any antinegative semiring and best possible over Boolean semirings if i = 1. Proof. Again we are going to show that the lower bound is best possible. Let S be a Boolean For each the matrices Ar = 0 ≤ r, s ≤ m, we consider r semiring. spair (r, s), s−1 + Es,1 J\ Ei,i , Bs = J \ Ei,i+1 if s < m and Bs = J \ E i,i+1 i=1 i=1

i=1

if s = m. Then

c(Ar ) = r(Ar ) = sr(Ar ) = sc(Ar ) = mr 1 (Ar ) = mc1 (Ar ) = r, c(Bs ) = r(Bs ) = sr(Bs ) = sc(Bs ) = mr 1 (Bs ) = mc1 (Bs ) = s, and Ar + Bs = J has row and column ranks equal to 1. Thus, these bounds are best possible over Boolean semirings. The rest of the proof follows directly from the deﬁnitions. Proposition 6.12. [45] Let S be a subsemiring in R+ . Then for A, B ∈ Mm,n (S), one has that c(A + B), r(A + B), sr(A + B), sc(A + B), mr 1 (A + B), mc1 (A + B) ≥ |ρ(A) − ρ(B)|. These bounds are exact and best possible. 6.2. Sylvester inequalities First, we show that the analog of the Sylvester lower bound does not hold for the factor rank. Example 6.13. Let S ⎡ 1 0 ⎢1 1 ⎢ ⎢ A = ⎢1 0 ⎢ .. .. ⎣. . 1 and B = AB = J.

At ,

0

be a Boolean semiring, ⎤ 0 ··· 0 0 ··· 0⎥ ⎥ ⎥ 1 ··· ⎥ .. . . .. ⎥ . . .⎦ 0

···

1

then f (A) = f (B) = n and f (AB) = 1 ≥ f (A) + f (B) − n = n since

However, we can prove the following.

Matrix Invariants over Semirings

27

Proposition 6.14. [45] Let S be an antinegative semiring, A ∈ Mm,n (S), B ∈ Mn,k (S). Then 1. f (AB) ≤ min{f (A), f (B)}, 0 if f (A) + f (B) ≤ n 2. f (AB) ≥ 1 if f (A) + f (B) > n, provided that S has no zero divisors. These bounds are exact, the upper bound is best possible, and the lower bound is best possible over Boolean semirings. The next example demonstrates that the standard lower bounds of the Sylvester inequality are not valid in the case S ⊆ R+ . Example 6.15. [45] Let us consider ⎡ ⎤ ⎡ 0 1 1 1 1 ⎢1 0 1 1⎥ ⎢1 ⎥ ⎢ A=⎢ ⎣ 1 1 0 4 ⎦ and B = ⎣ 0 1 1 4 0 0

1 1 0 0

0 0 1 1

⎤ 0 0⎥ ⎥, 1⎦ 1

then ρ(A) = 3, f (A) = 4, ρ(B) = f (B) = 2, and ⎡ ⎤ 1 1 2 2 ⎢1 1 2 2⎥ ⎥ AB = ⎢ ⎣ 2 2 4 4 ⎦, 2 2 4 4 so f (AB) = 1. Thus, 1 = f (AB) ≥ f (A)+f (B)−n = 4+2−4 = 2 and 6 = 4+2 = f (AI) + f (IB) ≤ f (AIB) + f (I) = 1 + 4 = 5. However, the following generalization for the lower bound is true. Proposition 6.16. [45] Let S ⊆ R+ , A ∈ Mm,n (S), B ∈ Mn,k (S). Then 0 if ρ(A) + ρ(B) ≤ n, f (AB) ≥ ρ(A) + ρ(B) − n if ρ(A) + ρ(B) > n. This bound is exact and best possible. Let us turn to the term rank now. Example 6.17. The inequality t(AB) ≤ min(t(A), t(B)) does not hold. For a counterexample, take A = C1 , B = R1 . Then t(AB) = t(Jn ) = n > 1. Also, there is no nontrivial multiplicative lower bound over an arbitrary semiring. Example 6.18. Set A = B = Jn over a ﬁeld whose characteristic is a divisor of n. Then t(AB) = t(nJn ) = 0.

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However, the following inequalities are true. Proposition 6.19. [45] Let S be an antinegative semiring. Then for any A ∈ Mm,n (S), B ∈ Mn,k (S), the inequalities t(AB) ≤ min(tr (A), tc (B)) and

t(AB) ≥

0 if t(A) + t(B) − n if

t(A) + t(B) ≤ n, t(A) + t(B) > n

hold. These are exact and best possible bounds. Proposition 6.20. [45] Let S be an antinegative semiring. For A ∈ Mm,n (S), B ∈ Mn,k (S), we have 0 ≤ z(AB) ≤ min{z(A) + z(B), k, m}. These bounds are exact and best possible for n > 2. Proposition 6.21. [45] Let S be an antinegative semiring without zero divisors. For O = A ∈ Mm,n (S), O = B ∈ Mn,k (S), and c(A) + r(B) > n, one has that 1 ≤ c(AB), r(AB), sr(AB), sc(AB), mr 1 (AB), mc1 (AB). These bounds are exact over any antinegative semiring without zero divisors and best possible over Boolean semirings. Proof. For O = A ∈ Mm,n (S) and O = B ∈ Mn,k (S), the matrix A has at least c(A) (sc(A), mc1 (A)) nonzero columns while B has at least r(B) (sr(B), mr 1 (B)) nonzero rows. Thus, if c(A) + r(B) > n, AB = O and hence these bounds are established. For the proof of exactness, let us take A = B = E1,1 . Let S be a Boolean semiring. For each pair (r, s), 1 ≤ r ≤ min{m, n}, 1 ≤ s ≤ min{k, n}, let us consider the matrices Ar =

r i=1

Ei,i +

m i=1

Ei,1 ,

Bs =

s i=1

Ei,i +

n

E1,i .

i=1

Then c(Ar ) = r(Ar ) = sr(Ar ) = sc(Ar ) = mr 1 (Ar ) = mc1 (Ar ) = r, c(Bs ) = r(Bs ) = sr(Bs ) = sc(Bs ) = mr 1 (Bs ) = mc1 (Bs ) = s by deﬁnition, and Ar Bs = J. Thus c(Ar Bs ) = r(Ar Bs ) = sr(Ar Bs ) = sc(Ar Bs ) = mr 1 (Ar Bs ) = mc1 (Ar Bs ) = 1. Note that the condition that c(A) + r(B) > n is necessary because for A = Ik ⊕ O and B = O ⊕ Ij , we have AB = O whenever k + j ≤ n.

Matrix Invariants over Semirings

29

Proposition 6.22. [45] Let S be a subsemiring in R+ . Then for A ∈ Mm,n (S) and B ∈ Mn,k (S), one has that c(AB), sc(AB), mc1 (AB) r(AB), sr(AB), mr 1 (AB)

≥

if

ρ(A) + ρ(B) ≤ n,

ρ(A) + ρ(B) − n if

ρ(A) + ρ(B) > n.

0

These bounds are exact and best possible. The following example, given in [51] for the spanning column rank, shows that standard analog for the upper bound of the rank of product of two matrices do not work for row and column ranks. ⎡ ⎤ 1 1 1 Example 6.23. [51] Let A = [3, 7, 7] ∈ M1,3 (Z+ ), B = ⎣ 0 1 1 ⎦. Then 0 0 1 c(A) = sc(A) = mc(A) = 2, c(B) = sc(B) = mc(B) = 3, and c(AB) = c(3, 10, 17) = sc(AB) = mc(AB) = 3. However, the following upper bounds are proved in [51]. Proposition 6.24. [51] Let S be an antinegative semiring. Then for A ∈ Mm,n (S) and B ∈ Mn,k (S), one has that c(AB) ≤ c(B), sc(AB) ≤ sc(B), mc1 (AB) ≤ mc1 (B), r(AB) ≤ r(A), sr(AB) ≤ sr(A), mr 1 (AB) ≤ mr 1 (A). These bounds are exact and best possible. Similarly, the following can be proved. Proposition 6.25. Let S be an antinegative semiring. Then for A ∈ Mm,n (S), B ∈ Mn,k (S), one has that mci (AB) ≤ mci (B), mr i (AB) ≤ mr i (A), i = 2, 3. These bounds are exact and best possible. Proposition 6.26. [42] Let S be a max-algebra then the following holds rk sym (AB) ≤ min{rk sym (A), rk sym (B)}. 6.3. Ranks of matrix union In [53, Lemma 3.17, Lemma 3.20], the following is proved.

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Proposition 6.27. Let S be an antinegative semiring. For any A ∈ Mm,n (S), and B ∈ Mm,k (S), the following inequalities for the rank of a matrix union hold. These inequalities are sharp and the upper bounds are best possible: max{f (A), f (B)} ≤ f (A|B) ≤ f (A) + f (B), max{t(A), t(B)} ≤ t(A|B) ≤ t(A) + t(B). Proposition 6.28. Let S be an antinegative semiring. Then for any A ∈ Mm,n (S), B ∈ Mm,k (S), it holds that max{r(A), r(B)} ≤ r(A|B). Proof. This is straightforward by the deﬁnition of row rank.

Let us show that the row rank of the matrix union (A|B) can be bigger than the sum of the row ranks of A and B. Example 6.29. Let us consider the following two matrices ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 0 ⎢0 0 1⎥ ⎢0 0 0⎥ ⎥ ⎢ ⎥ A=⎢ ⎣ 0 0 1 ⎦, B = ⎣ 1 0 0 ⎦. 0 1 0 1 0 0 Then it is straightforward to see that r(A) = 2, r(B) = 1. Note that by Example 4.4, it holds that r(A|B) = 4 > 3 = r(A) + r(B). Proposition 6.30. Let S be an antinegative semiring. Then for any A ∈ Mm,n (S), B ∈ Mm,k (S), it holds that c(A|B) ≤ c(A) + c(B). Proof. This follows directly from the deﬁnition of column rank.

The following example shows that the column rank of the matrix union (A|B) can be less than the minimum of column ranks of A and B. Example 6.31. Let us consider the matrices ⎤ ⎡ ⎡ 1 0 1 0 0 0 A = ⎣ 0 1 0 1 ⎦, B = ⎣ 0 1 0 0 1 1 1 0

1 0 1

⎤ 1 1 ⎦. 0

Then one has by Example 4.4 that c(A) = 4, c(B) = 4, but c(A|B) = 3 < 4. For the other row and column ranks, the situation is analogous. Propositions 6.32 and 6.33 follow directly from the deﬁnitions of bi-determinant and tropical singularity, respectively.

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31

Proposition 6.32. Let S be an antinegative semiring. For any A ∈ Mm,n (S), B ∈ Mm,k (S), the following inequalities for the rank of matrix union hold. These inequalities are sharp and the upper bound is best possible: max{rk sym (A), rk sym (B)} ≤ rk sym (A|B) ≤ rk sym (A) + rk sym (B). Proposition 6.33. Let S be an antinegative semiring. For any A ∈ Mm,n (S), B ∈ Mm,k (S), the following inequalities for the rank of matrix union hold. These inequalities are sharp and the upper bound is best possible: max{trop(A), trop(B) ≤ trop(A|B) ≤ trop(A) + trop(B).

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Quadratic Forms Kazimierz Szymiczek University of Silesia, Katowice, Poland E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quadratic forms over ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Witt rings of ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Prime ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Nilradical, torsion, divisors of zero, and units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Hasse and Witt invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Milnor’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Equivalence of quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Witt equivalence of ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Fields with ﬁnite square class number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Witt equivalence of global ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Function ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Selected applications of function ﬁelds techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. u-invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Sums of squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Pythagoras number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Witt rings of rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Witt rings of commutative rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Witt rings of Dedekind domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Witt rings of skew ﬁelds and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. More general Witt rings and groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Abstract Witt rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Elementary type conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Reduced theory of quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Historical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00202-7 35

36 37 39 41 41 43 45 47 47 49 50 51 53 53 56 58 58 59 60 61 61 63 64 64 65 65 67 69 71

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1. Introduction Quadratic forms constitute a large domain of research, with roots in classical mathematics and remarkable developments in the last decades. Its origins go back to Fermat and Euler, and at the times of Gauss there existed already a deep theory of quadratic forms with integer coefﬁcients. A new stimulus was provided by two of the Hilbert problems (11th and 17th) on quadratic forms resolved in the 1920s by Hasse and Artin. Together with the important 1937 paper by Witt [240], this laid the foundation for the development of the algebraic theory of quadratic forms with the leading role played by the theory of quadratic forms over ﬁelds of characteristic different from two. This survey concentrates on that motor force behind the spectacular developments we have witnessed lately in the subject of quadratic forms. Sections 2–4 contain introductory material. Here, we set up the notation, terminology, recall the fundamentals of the theory, and provide motivation for the material discussed in the next sections. In Section 5, we discuss brieﬂy the Milnor Conjecture and its solution by Voevodsky. In Section 6, the classiﬁcation results for quadratic forms and Witt rings are presented. While classiﬁcation of quadratic forms up to equivalence is an obvious central problem of the theory, its solution does not sufﬁce to determine the structure of the Witt ring. We consider the classiﬁcation of Witt rings over a ﬁeld F to be the ultimate goal of the theory and report on the results over those ﬁelds where the theory is complete. Section 7 discusses selected applications of function ﬁelds of quadratic forms. This technique brings to quadratic form theory the methods of algebraic geometry, which proved to be extremely important and successful. The subsection on the u−invariant shows the strength of the function ﬁelds compared with the earlier more elementary and direct approach. Section 8 discusses the properties of particular quadratic forms, the sums of squares. Here, three most important themes are brought up which are as follows: the level of a ﬁeld or ring, the composition of sums of squares, and the Pythagoras number of ﬁelds and rings. In Section 9, we brieﬂy discuss the most natural generalization of the original Witt ring, the Witt ring of a commutative ring. As a vehicle, we have chosen two problems. First, for an integral domain R and its ﬁeld of fractions F , we report on the results on the natural Witt ring homomorphism W(R) → W(F ). Second, for a Dedekind domain R, there is an exact Knebusch–Milnor sequence i ∂ 0 → W(R) −→ W(F ) −→ W(R/p), p

and the problem we discuss is the computation of the cokernel of ∂. In Section 10, we describe the abstract approaches to quadratic form theory that emerged from the work on the classiﬁcation of Witt rings and from the attempts to generalize Witt rings to the context of semilocal rings. We explain the most important unsolved problem of the abstract theory, the elementary type conjecture. Its solution would give a full classiﬁcation of Witt rings of ﬁelds with ﬁnite number of square classes. We also present the basic result on the structure of reduced Witt rings.

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In Section 11, we sketch the developments leading to the Witt’s 1937 paper, beginning with Minkowski and Hilbert. While the literature on algebraic theory of quadratic forms is huge and grows steadily, there are several books synthesizing the work in the area. Of these, we mention only three: Scharlau [204] covering Hermitian forms with an extensive literature up to 1982; Lam [143], covering almost all areas of quadratic form theory over ﬁelds of characteristic = 2; and Elman, Karpenko, and Merkurjev [65], a new important book written from the viewpoint of algebraic geometry (not available at the time of writing this chapter). In this survey, we do not discuss the areas of research that properly belong to quadratic form theory over domains other than ﬁelds, such as • classical theory of quadratic forms over rings of integers in an algebraic number ﬁeld, especially over Z, for this see [11, 37, 83, 114, 125, 174, 181], and the literature cited there in, • quadratic forms over algebraic geometric varieties, see [3, 7, 33, 120, 180, 186, 187, 223] • quadratic and Hermitian forms over general rings, polynomial rings, and afﬁne algebras [126, 204]. We also do not discuss the areas of research close to quadratic form theory and naturally related and motivated by the developments in quadratic form theory, such as • • • •

ordered ﬁelds, preorderings, and valuation theory [141, 142] central simple algebras, linear algebraic groups [107] Galois theory [1, 82, 104, 105, 140, 175, 176] real commutative algebra, semialgebraic geometry, and algebraic topology [63, 106, 163, 198, 207] • higher level orderings of ﬁelds, Witt rings of higher level [17, 20, 21, 185] • forms of higher degree, higher levels, and higher Pythagoras numbers [18, 29, 85, 166, 203]. 2. Quadratic forms over ﬁelds In this section, we consider quadratic forms and bilinear forms over ﬁelds of characteristic different from 2. Given a ﬁnite dimensional vector space V over the ﬁeld F , a quadratic form q on V is a function q : V → F such that the associated function bq : V × V → F deﬁned by bq (u, v) = 12 (q(u + v) − q(u) − q(v)) is bilinear and q(av) = a2 q(v) for all a ∈ F and v ∈ V . The pair (V, q) is then said to be a quadratic space and (V, bq ) a bilinear space over the ﬁeld F . Two vectors u, v ∈ V are said to be orthogonal if bq (u, v) = 0. Two quadratic spaces (V1 , q1 ) and (V2 , q2 ) over F are said to be isometric when there is a vector space isomorphism ϕ : V1 → V2 satisfying q2 (ϕ(v)) = q1 (v) for all v ∈ V1 . The two quadratic forms q1 and q2 are then said to be equivalent, written q1 ∼ = q2 . For a quadratic form q on V , the set of nonzero values of the form q is

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denoted as DF (q). The elements in DF (q) are said to be represented by q over F . Since q(av) = a2 q(v) for a ∈ F , it follows that the set DF (q) consists of whole cosets of the multiplicative group F˙ modulo the subgroup F˙ 2 of squares. Hence, DF (q) can be viewed as a subset of the group F˙ /F˙ 2 of square classes of F . For a quadratic space (V, q), the dimension of V is said to be the dimension of the quadratic form q, written as dim q. If B = {v1 , . . . , vn } is a basis for V , the matrix B = [bq (vi , vj )] is said to be the matrix of q with respect to B. Distinct bases B1 and B2 produce congruent matrices B1 and B2 = PB1 P T , where P is a nonsingular matrix. Hence, det B1 and det B2 , if nonzero, lie in the same coset (det B)F˙ 2 , which is said to be the determinant of the form q (or of the space (V, q)), written as det q. If det B = 0 for some basis B, we set det q = 0. Forms with nonzero determinant are called nonsingular. This is equivalent to the fact that the map V → V ∗,

v → bq ( , v)

is an isomorphism of V onto the dual space V ∗ . Each quadratic form q over a ﬁeld of characteristic not 2 can be diagonalized. That is, there is a basis B for V such that the matrix of q with respect to B is a diagonal matrix. In this case, B consists of pairwise orthogonal vectors. We often identify quadratic forms with their diagonal matrices. If (V1 , q1 ) and (V2 , q2 ) are quadratic spaces, then so is (V1 ⊕ V2 , q1 ⊥ q2 ), where (q1 ⊥ q2 )(v1 , v2 ) = q1 (v1 ) + q2 (v2 ). This is called the orthogonal direct sum of forms q1 and q2 . Also, the tensor product V1 ⊗ V2 can be equipped with the structure of a quadratic space (V1 ⊗ V2 , q), where the associated bilinear form bq equals the tensor product bq1 ⊗ bq2 . Hence, bq (v1 ⊗ v2 , v1 ⊗ v2 ) = bq1 (v1 , v1 ) · bq2 (v2 , v2 ) for all simple tensors v1 ⊗ v2 , v1 ⊗ v2 in V1 ⊗ V2 . The form q is then called the tensor product of the quadratic forms q1 and q2 , written as q1 ⊗ q2 . Direct orthogonal sum and tensor product of nonsingular quadratic forms are likewise nonsingular. If q1 = (a1 , . . . , an ) and q2 = (b1 , . . . , bm ) are diagonalized forms, then q1 ⊥ q2 = (a1 , . . . , an , b1 , . . . , bm ), q1 ⊗ q2 = (a1 b1 , . . . , an b1 , a2 b1 , . . . , an bm ). The very useful Witt cancellation theorem states that if q is nonsingular and q ⊥ q1 ∼ = q ⊥ q2 , then q1 ∼ = q2 . A quadratic form (1, a), a ∈ F˙ , is called one-fold Pﬁster form, and an n-fold tensor product (1, a1 ) ⊗ · · · ⊗ (1, an ) of one-fold Pﬁster form is called an n-fold Pﬁster form. A quadratic form q is said to be isotropic if there exists a nonzero vector v ∈ V so that q(v) = 0. A simple but fundamental example of a nonsingular isotropic form is the hyperbolic plane. This is the two-dimensional form h with diagonalization (1, −1) in some basis of the plane. If a quadratic form q is isotropic, then it splits off a hyperbolic plane h1 , i.e., q ∼ = h1 ⊥ q1 for some quadratic form q1 . Continuing with q1 , we ultimately obtain a decomposition q∼ = h1 ⊥ · · · ⊥ hi ⊥ qa , where h1 , . . . , hi are hyperbolic planes and qa is anisotropic. By Witt cancellation, the form qa , called the anisotropic part of q, is unique up to isometry, and also, the

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number i (called the Witt index of q) is unique. If qa = 0, the form q is said to be hyperbolic. It is an important property of Pﬁster forms that if a Pﬁster form is isotropic, then it is necessarily hyperbolic. Another fundamental property is that for a Pﬁster form q, the value set DF (q) is a group under multiplication. This implies immediately that the level of a nonformally real ﬁeld is always a power of two (see Section 8.1). Here, the level s(A) of a ring A is the minimal number of summands in a representation of −1 ∈ A as a sum of squares of elements in A (or ∞ if such a representation does not exist). Two quadratic forms q and g over F are said to be similar (or Witt equivalent), written as q ∼ g, if their anisotropic parts are isometric, qa ∼ = ga . An easy observation is that for quadratic forms q and g over F , dim q = dim g

and q ∼ g ⇒ q ∼ = g.

(2.1)

Two fundamental problems of quadratic form theory are the representation problem and the classiﬁcation problem. The representation problem over a ﬁeld F asks for an explicit determination of the elements represented by any quadratic form over F . The classiﬁcation problem over a ﬁeld F asks for criteria for deciding whether any given quadratic forms over F are equivalent or not. If the representation problem is solved over a ﬁeld F , the equivalence problem can be solved by induction on dimension. Indeed, for two forms q and g of the same dimension n, take any element a ∈ DF (q) and check whether a ∈ DF (g) or not. If a is not represented by g, the forms q and g are not equivalent. If a is represented by g, the forms q and g can be diagonalized with the element a occurring on the diagonal. Hence, q = (a) ⊥ q1 and g = (a) ⊥ g1 for some (n − 1)-dimensional forms q1 , g1 , and by the Witt cancellation theorem, q ∼ = g ⇐⇒ q1 ∼ = g1 . Thus, the question is reduced to a lower dimension. Another elementary observation is that the representation problem is equivalent to the isotropy problem: to decide whether or not a given quadratic form is isotropic. For a ∈ DF (q) ⇐⇒ q ⊥ (−a) is isotropic.

3. Witt rings of ﬁelds We continue to assume that F is a ﬁeld of characteristic different from 2. For a nonsingular quadratic form q over F , we write q for the class of quadratic forms similar to q. This is the Witt class of q. Witt classes of nonsingular quadratic forms over F with addition and multiplication induced by direct orthogonal sum and tensor product form a commutative ring called the Witt ring of the ﬁeld F and denoted as W(F ).1 When the characteristic of F equals 2, a similar construction produces the Witt ring of nonsingular bilinear forms over F , also denoted as W(F ). Below we refer to this ring when we allow characteristic 2. 1 The same notation, W(A), is used for the ring of big Witt vectors over a ring or ﬁeld A, a totally different

concept. See Michiel Hazewinkel, Witt vectors. Part 1, this volume.

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The Witt ring has the following description in terms of generators and relations. Consider the integral group ring Z[F˙ /F˙ 2 ] of the group of square classes of the ﬁeld F . Then, W(F ) = Z[F˙ /F˙ 2 ]/J, where J is the ideal in the group ring Z[F˙ /F˙ 2 ] generated by the set, {[a] + [b] − [c] − [d] : (a, b) ∼ = (c, d)} ∪ {[1] + [−1]} . This is a straightforward consequence of the Witt theorem on chain equivalence of quadratic forms (see [240, Satz 7]). For a Witt class q, we deﬁne the dimension-index eq = dim q

(mod 2) ∈ Z/2Z

and the discriminant dq = (−1) 2 n(n−1) det q ∈ F˙ /F˙ 2 , 1

where n = dim q. These are well-deﬁned invariants of the similarity class. Moreover, e : W(F ) → Z/2Z is a ring epimorphism. Its kernel consists of the Witt classes q with even-dimensional q and is said to be the fundamental ideal of the Witt ring W(F ), written as I(F ). Further, d : W(F ) → F˙ /F˙ 2 is a well-deﬁned map, but it fails to be a homomorphism of the additive group W(F ) (for instance, take F = R). However, if we restrict the discriminant map to the fundamental ideal I(F ), we obtain a surjective group homomorphism d : I(F ) → F˙ /F˙ 2 of the additive group I(F ) onto the group of square classes. Interestingly enough, the kernel of this homomorphism equals I 2 (F ), the additive group of the square of the fundamental ideal. This prompts looking at higher powers I n (F ) of the fundamental ideal. An elementary observation is that I n (F ) is generated as an Abelian group by the Witt classes of all n-fold Pﬁster forms over F . Much deeper and difﬁcult to prove is the following theorem known as the Hauptsatz. It was proved by Arason and Pﬁster [5] in 1971. Theorem 3.1 (Arason–Pﬁster Hauptsatz). For a positive-dimensional anisotropic quadratic form q over a ﬁeld F , if q ∈ I n (F ), then dim q ≥ 2n . As a consequence, we get the intersection property in the Witt ring W(F ) of the ﬁeld F , ∞ n=0

I n (F ) = 0.

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3.1. Prime ideals Prime ideals of the Witt ring are described as follows (Harrison [84], Lorenz and Leicht [159]). Theorem 3.2. Let F be a ﬁeld of characteristic not 2. (a) If the Witt ring W(F ) has a prime ideal p = I(F ), then the ﬁeld F is formally real and the set P := { a ∈ F˙ : 1, −a ∈ p } deﬁnes an ordering of the ﬁeld F. (b) Let F be a formally real ﬁeld and let P be an ordering of F. Let p0 be the ideal of the Witt ring W(F ) generated by the set { 1, −a ∈ W(F ) : a ∈ P }. Then, p0 is a minimal prime ideal of the Witt ring W(F ). Moreover, p0 ⊂ I(F ) and p0 = I(F ). Hence, MinSpec W(F ) = {I(F )} is a singleton set when the ﬁeld F is nonreal, and MinSpec W(F ) contains at least one nonmaximal minimal prime ideal when the ﬁeld F is formally real. Each ordering P of a formally real ﬁeld F gives rise to a signature homomorphism sgnP : W(F ) → Z sending the class q to the signature of the form q for the ordering P. The map σ : X(F ) → MinSpec W(F ), P → ker sgnP is a bijective correspondence between the set X(F ) of all orderings of the ﬁeld F and the minimal prime ideals of the Witt ring W(F ). The inverse map for σ is the map π deﬁned as follows: π : MinSpec W(F ) → X(F ), p → P = {a ∈ F˙ : 1, −a ∈ p}. The set X(F ) can be given the induced Zariski topology from the prime spectrum and this turns X(F ) into a Boolean space (compact, Hausdorff, and totally disconnected). Craven [42] proved that every Boolean space can be obtained as the space of orderings of some ﬁeld. For a realization of Boolean spaces as spaces of orderings of higher level, see Osiak [183, 184]. 3.2. Nilradical, torsion, divisors of zero, and units For a nilpotent element x ∈ W(F ), we obviously have e(x) = 0 and hence Nil W(F ) ⊆ I(F ). When F is non-formally real (nonreal, for short) ﬁeld, we actually have Nil W(F ) = I(F ). When F is formally real, we observe that nilpotent elements

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in W(F ) have zero signatures at every ordering of F , and so the nilradical Nil W(F ) is contained in the intersection of the kernels of all signature homomorphisms. Since the latter are all minimal prime ideals in W(F ), their intersection actually equals the nilradical Nil W(F ). For a nonreal ﬁeld F of level s, the unit element 1 ∈ W(F ) has ﬁnite order 2s in the additive group W(F ) and so 2sW(F ) = 0. This is an immediate consequence of the theory of Pﬁster forms. Thus, W(F ) is torsion group and every element is two-primary torsion. When F is formally real, torsion elements have zero signatures at all orderings of the ﬁeld, so it follows that Tors W(F ) ⊆ Nil W(F ). A deeper result says that, in fact, for every formally real ﬁeld F, Tors W(F ) = Nil W(F ). This is a consequence of the local–global principle for torsion elements of Witt rings ﬁrst proved by Pﬁster [191]. Theorem 3.3 (Pﬁster local–global principle). Let F be a formally real ﬁeld and let q be an anisotropic quadratic form over F. For every ordering P of the ﬁeld F , choose a real closure FP inducing the ordering P on F. The following statements are equivalent. (a) q is a torsion element of the Witt ring W(F ). (b) qFP = 0 in the Witt ring W(FP ), for every ordering P of the ﬁeld F. (c) sgnP q = 0 for all orderings P of the ﬁeld F . Moreover, every torsion element in W(F ) is two-primary torsion. A ﬁeld F is said to be Pythagorean when every sum of squares of nonzero elements of F is a square of a nonzero element in F . Thus, a Pythagorean ﬁeld is automatically formally real. For all ﬁelds F except for Pythagorean ﬁelds, the set ZD(W(F )) of zero divisors of the Witt ring W(F ) coincides with the ideal I(F ). And for a Pythagorean ﬁeld F , an element of the Witt ring of F is a zero divisor in W(F ) if and only if it lies in a minimal prime ideal of the ring W(F ). Hence for a Pythagorean ﬁeld F , ZD(W(F )) = {ker sgnP : P ∈ X(F )}. The following theorem gives an important characterization of Pythagorean ﬁelds in terms of their Witt rings. Theorem 3.4. For a formally real ﬁeld F , F

is Pythagorean ⇐⇒ Tors W(F ) = 0 ⇐⇒ Nil W(F ) = 0.

This theorem implies a solution of the classiﬁcation problem for quadratic forms over Pythagorean ﬁelds. If q and g are nonsingular quadratic forms over a Pythagorean ﬁeld F , then q ∼ = g if and only if dim q = dim g and sgnP q = sgnP g for every ordering P of the ﬁeld F. The units U(W(F )) of the Witt ring W(F ) can be determined as follows.

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Theorem 3.5. If the ﬁeld F is nonreal, then U(W(F )) = 1 + I(F ). If F is formally real, then q ∈ U(W(F )) if and only if sgnP q = ±1 for all orderings P of F . Finally, for any ﬁeld F , 0 and 1 are the only idempotents of the Witt ring W(F ). The proofs of the results in this and preceding sections can be found in any standard text on algebraic theory of quadratic forms (for instance, [143, 204, 228]). 4. Hasse and Witt invariants For a nonsingular diagonal quadratic form q = (a1 , . . . , an ) with the entries ai in a ﬁeld F of characteristic = 2, we deﬁne the Hasse algebra H(q) of the form q as the following tensor product of quaternion algebras: ai , aj H(q) := . F 1≤i≤j≤n

Here, for a, b ∈ F˙ , the symbol ( a,b F ) denotes the quaternion algebra over F with structure constants a, b. This is a four-dimensional F −algebra with a basis 1, i, j, k and the multiplication rules i2 = a, j 2 = b, ij = k = −ji. If q = (a1 , . . . , an ) and g = (b1 , . . . , bn ) are equivalent quadratic forms, then H(q) ∼ = H(g). In other words, the Hasse algebra of a quadratic form q is uniquely determined up to algebra isomorphism and is an equivalence invariant. To calculate the Hasse algebra one uses the following observation det q, det g ∼ ⊗ M (F ), H(q ⊥ g) = H(q) ⊗ H(g) ⊗ (4.1) F where is an appropriate positive integer. The formula (4.1) suggests that it is suitable to pass from the Hasse algebra to the similarity class of the algebra in the Brauer group Br(F ) of the ﬁeld F . For a nonsingular quadratic form q = (a1 , . . . , an ) over a ﬁeld F of characteristic = 2, the Hasse invariant h(q) of the form q is deﬁned to be the similarity class of the Hasse algebra H(q) of the form q in the Brauer group of the ﬁeld F :

[ai , aj ] ∈ Br(F ). h(q) := [H(q)] = 1≤i≤j≤n

Here [ai , aj ] stands for the similarity class of the quaternion algebra Brauer group of the ﬁeld F .

ai ,aj F

in the

Theorem 4.1 (Classiﬁcation theorem for quadratic forms of dimension ≤ 3). Let q and g be nonsingular quadratic forms over a ﬁeld F of characteristic = 2 and let dim q ≤ 3 and dim g ≤ 3. Then, we have q∼ =g

⇔

dim q = dim g, det q = det g, h(q) = h(g).

44

K. Szymiczek

The Hasse invariant, like the determinant, is an invariant for equivalence but not for similarity of quadratic forms. However, as in the case of the determinant, one can correct this by multiplying the Hasse invariant with a suitable factor. This works in all dimensions, but we shall restrict attention to even-dimensional forms. We deﬁne w(q) := h(q) · [−1, −1]n(n+1)/2 ,

where dim q = 2n.

(4.2)

It turns out that the new function is a similarity invariant on the set of even-dimensional forms with discriminant 1 ∈ F˙ /F˙ 2 and this set is precisely I 2 (F ). Hence for q ∈ I 2 (F ), we set w(q) := w(q) ∈ Br(F ), and call it the Witt invariant of the form q, and of the class q ∈ I 2 (F ). Thus,

h(q) · [−1, −1] when 2n ≡ 2, 4 mod 8, w(q) = h(q) when 2n ≡ 0, 6 mod 8. Notice that w(q) ∈ Quat F , the subgroup of Br(F ) generated by classes of quaternion algebras. Since w(x, −ax, −bx, abx) = w(1, −a, −b, ab) = [a, b], w((1, −a) ⊗ (1, −b) ⊗ (1, −c)) = 1, for all x, a, b, c ∈ F˙ , it easily follows that the map w : I 2 (F ) → Quat F is a group epimorphism and I 3 (F ) ⊆ ker w. It has been of central importance for quadratic form theory whether actually I 3 (F ) = ker w. This was proved for various classes of ﬁelds F including local and global ﬁelds and ﬁelds of transcendence degree ≤ 1 over a real closed ﬁeld. A more general approach presented in [69] showed that I 3 (F ) = ker w for all 1amenable ﬁelds. Here√ F is said to√be 1-amenable if for any multiquadratic extension K of F , i.e. K = F( a1 , · · · , ar ), ai ∈ F˙ , the kernel ker(W(F ) → W(K)) of the natural ring homomorphism W(F ) → W(K) is generated as an ideal by the onefold Pﬁster forms 1, −ai . A ﬁeld extension K/F is said to be excellent if, for any quadratic form q over F , the anisotropic part of q ⊗ K is deﬁned over F . F is 1amenable if every multiquadratic extension is excellent. Excellence and amenability were investigated in [70, 71], but then the ﬁrst examples of ﬁelds, which are not 1amenable were found in [72], thus showing that I 3 (F ) = ker w cannot be deduced in general from 1-amenability. Further, counterexamples to amenability and excellence were given in [149, 210–212]. The question whether I 3 (F ) = ker w was resolved by Merkurjev [168] in 1981 as a part of an even more spectacular result discussed below. We shall use the Milnor K-theory group K2 F and its factor group, the elementary Abelian two-group k2 F := K2 F/2K2 F generated by the cosets of symbols {a, b} with a, b ∈ F˙ . The correspondence {a, b} → ( a,b F ) induces a group homomorphism h : k2 F → Br 2 (F ), where Br 2 (F ) is the subgroup of the Brauer group Br(F ) consisting of elements of order ≤ 2. However, the correspondence {a, b} → 1, −a, −b, ab

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45

induces a group homomorphism s : k2 F → I 2 (F )/I 3 (F ), and the Witt invariant induces a homomorphism e2 : I 2 (F )/I 3 (F ) → Quat F ⊆ Br 2 (F ). These homomorphisms yield the commutative diagram k2 F s I 2 (F )/I 3 (F )

h e2

Br2 (F )

It was proved by Milnor [173] that s is an isomorphism. Merkurjev’s contribution is the following important result. Theorem 4.2 (Merkurjev [168]). The homomorphism h is an isomorphism for all ﬁelds F of characteristic = 2. It follows that the homomorphism e2 is also an isomorphism and consequently I 3 (F ) = ker w. Thus, we get a complete characterization of quadratic forms in I 3 (F ): these are even dimensional forms with trivial discriminant and trivial Witt invariant. Another consequence is an improvement on the classiﬁcation theorem 4.1 contained in Theorem 6.2 below. Merkurjev’s theorem shows that Quat F = h(k2 F ) = Br 2 (F ), which solves an old problem in the theory of algebras (any element of order 2 in the Brauer group B(F ) is expressible as a product of classes of quaternion algebras, or, equivalently, any central division algebra with an involution over F is similar to a tensor product of quaternion algebras). Merkurjev’s proof that h : k2 F → Br 2 (F ) is an isomorphism depended on a result of Suslin on the quadratic norm residue symbol available at that time only through deep results in Quillen’s K-theory. New proofs were given subsequently by Arason [4], Wadsworth [238], and Merkurjev [171] and they do not depend on Quillen’s K-theory. While the setup of Merkurjev’s theorem can be generalized to commutative rings, there is, however, no analog to Merkurjev’s theorem. Parimala and Sridharan have given in [188] an example of a commutative ring R for which the Clifford algebra map (corresponding to the Witt invariant w) from strictly even rank quadratic forms over R to Br 2 (R) is not surjective. 5. Milnor’s Conjecture In a paper published in 1970, Milnor [173] observed interrelations among Witt rings, K-theory, and Galois cohomology. For a ﬁeld F of characteristic not 2, the K-theory groups are Milnor’s Kn F and the factor groups kn F := Kn F/2Kn F , and the cohomology groups are the Galois cohomology H n (F ) = H n (Gal (F /F ), Z/2Z), where F denotes a separable closure of F. Milnor deﬁned homomorphisms sn : kn F → I n (F )/I n+1 (F )

46

K. Szymiczek

induced by the map sending the pure symbol {a1 , . . . , an } ∈ Kn F onto the Witt class of the Pﬁster form (1, −a1 ) ⊗ · · · ⊗ (1, −an ) ∈ I n (F ). Since I n (F ) is additively generated by all n-fold Pﬁster forms, every sn is surjective, and Milnor proved that s1 and s2 are bijective. He also deﬁned homomorphisms hn : kn F → H n (F ) induced by the map sending the pure symbol {a1 , . . . , an } ∈ Kn F onto the cup product (a1 ) ∪ · · · ∪ (an ) ∈ H n (F ). For n = 0, 1, 2, the groups H n (F ) can be identiﬁed with Z/2Z, F˙ /F˙ 2 , Br 2 F , and dimension index, discriminant, and Witt invariant can be viewed as surjective group homomorphisms en : I n (F ) → H n (F ) satisfying ker en = I n+1 (F ) (for n = 2, this assumes Merkurjev’s theorem). Hence in these cases, we have isomorphisms en : I n (F )/I n+1 (F ) → H n (F ). It was an open problem whether such isomorphisms exist for n ≥ 3. The existence of these isomorphisms is of importance for the classiﬁcation of quadratic forms, and so it is highly desirable to extend these existence results for other values of n. In 1975, Arason [2] proved the existence of the group homomorphism e3 , and in 1989 Jacob and Rost [103] and independently Szyjewski [222] proved the existence of e4 . The homomorphisms sn , hn , and the supposed maps en combine into the diagram kn F sn I n (F )/I n+1 (F )

hn en

H n (F )

The Milnor conjecture asserts that sn and hn are isomorphisms for all n and all ﬁelds F of characteristic not 2. Thus, the difﬁcult and, in fact, intractable question in quadratic form theory about the existence of the invariants en has been transferred to K-theory and cohomology theory. The bijectivity of hn for all n was proved by Voevodsky in 1996 and the bijectivity of sn shortly thereafter by Orlov, Vishik and Voevodsky. Hence, for any n and any ﬁeld F of characteristic not 2, the map en = hn ◦ sn−1 : I n (F )/I n+1 (F ) → H n (F ) is an isomorphism sending the cosets (1, −a1 ) ⊗ · · · ⊗ (1, −an ) + I n+1 (F ) of Pﬁster forms onto the cup products (a1 ) ∪ · · · ∪ (an ) ∈ H n (F ).

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47

The ﬁnal versions of their papers are now available [182, 236]. There are also several accounts of Voevodsky’s work explaining with various degree of detail the proofs of Milnor’s conjecture (see, Morel [179]). Therefore, we shall not discuss here the papers [182, 236]. We refer to the survey by Pﬁster [199] and to the paper by Arason and Elman [6] for consequences of Voevodsky’s results in quadratic form theory. We shall mention a few newer developments in the following sections. 6. Classiﬁcation The classiﬁcation of quadratic forms up to equivalence may be considered the central problem of quadratic form theory. By (2.1), this is essentially the same as classiﬁcation of quadratic forms up to similarity. Hence, solving the classiﬁcation problem for forms over a ﬁeld F , we determine the elements of the Witt ring W(F ). However, this says nothing about the ring structure of W(F ). Hence, it is natural to split the classiﬁcation problem into two parts. First, classify quadratic forms up to equivalence, and second, classify the Witt rings of ﬁelds up to isomorphism. 6.1. Equivalence of quadratic forms The classical invariants of quadratic forms are dimension, determinant, Hasse invariant, and the total signature. The latter is the totality of all signatures sgn P deﬁned by all orderings P of the ﬁeld in question. Total signature occurs only for formally real ﬁelds. The class of ﬁelds for which classical invariants classify quadratic forms was characterized by Elman and Lam [68] as follows. Theorem 6.1. Quadratic forms over a ﬁeld F are classiﬁed by their dimension, determinant, Hasse invariant, and total signature if and only if the ideal I 3 (F ) of the Witt ring W(F ) is torsion-free. When the ﬁeld F is nonreal, the Witt ring W(F ) is the torsion so that torsion-freeness of I 3 (F ) is to be interpreted as I 3 (F ) = 0. Observe that, since I 3 (F ) is torsion-free for all local and global ﬁelds, this theorem recaptures the classical results of Hasse on the classiﬁcation of quadratic forms over local and global ﬁelds (see Section 11 for Hasse’s results). Theorem 6.1 was proved before Merkurjev’s Theorem 4.2. Using Merkurjev’s theorem, the proof would become much easier. A reﬁned version of the proof not using Merkurjev’s theorem is given in Lam’s book [143, pp. 440–443]. For more general classiﬁcation results involving the classical invariants, see [16] (Hermitian forms over central simple algebras with involution) and [158] (classiﬁcation of central simple algebras with involution). Voevodsky’s theorem gives new possibilities for the classiﬁcation of quadratic forms. When we want to check whether two quadratic forms q, g over a ﬁeld F of characteristic different from 2 are equivalent, we may assume that dim q = dim g = n and consider the class ϕ = q ⊥ −g. From (2.1), we have q ∼ = g ⇐⇒ ϕ = 0 ∈ W(F ).

48

K. Szymiczek

We shall write en for the composition I n (F ) → I n (F )/I n+1 (F ) → H n (F ) and view the values of the homomorphisms en as invariants of quadratic forms in I n (F ). In order to compute en (ϕ ), we need to know that ϕ ∈ I n (F ). The latter is equivalent to requiring that ei (ϕ ) = 0 for i = 0, 1, . . . , n − 1. Theorem 6.2 (General classiﬁcation theorem). Let q, g be quadratic forms over a ﬁeld F of characteristic different from 2 and assume that dim q = dim g = n. Set ϕ = q ⊥ −g, and let k satisfy 2k ≤ 2n < 2k+1 . If ei (ϕ ) = 0 for some i ≤ k, then q and g are not equivalent forms. If ei (ϕ ) = 0 for i = 1, . . . , k, then q and g are equivalent forms. Proof. The class ϕ is even-dimensional and hence lies in I(F ) and so e0 (ϕ ) = 0. If e1 (ϕ ) = 0, we have ϕ ∈ I 2 (F ), and if is the smallest index for which e (ϕ ) = 0, then necessarily ϕ = 0 ∈ W(F ) and so q and g are inequivalent. If ei (ϕ ) = 0 for i = 1, . . . , k, then by Voevodsky’s theorem ϕ ∈ I k+1 (F ). However, dim ϕ = 2n < 2k+1 implies ϕ = 0 ∈ W(F ) by the Arason–Pﬁster Hauptsatz, but ϕ = 0 ∈ W(F ) and dim q = dim g imply q ∼ = g. 6.1.1. Trace forms For a ﬁnite separable ﬁeld extension K/F , the trace functional tr K/F : K → F deﬁnes a symmetric bilinear form K × K → F,

(x, y) → tr K/F (xy).

The associated quadratic form is called the trace form and denoted tK/F . When F is a number ﬁeld, trace forms tK/F are quadratic forms over F and so are classiﬁed up to isometry by the four classical invariants. Here, dim tK/F = [K : F ], det tK/F = NK/F (f (θ)), where K = F(θ) and f is the minimal polynomial of θ over F . The signature sgnP tK/F with respect to an ordering P of F equals the number of the extensions of P to orderings of K. In particular, we always have sgnP tK/F ≥ 0. The details may be found in [34]. The Hasse invariant of tK/F was computed by Serre [208] (see also [34, Chapter II]). A quadratic form q (the Witt class q ∈ W(F )) over a number ﬁeld F is said to be positive if the form (the Witt class) has nonnegative signature with respect to every ordering of F . If F has no orderings, then every quadratic form (and every Witt class) over F is positive. As mentioned above, trace forms (and their Witt classes) are positive. Conner and Perlis [34] proved that in the Witt ring W(Q) the converse holds: if a Witt class X ∈ W(Q) is positive, then there is a ﬁnite extension K of Q such that tK/Q = X. For an arbitrary number ﬁeld F as a base ﬁeld the result, was proved by Scharlau [205]. A characterization of positive quadratic forms up to isometry was found by Krüskemper [131]. He proved that every positive quadratic form of even degree n ≥ 4 over a number ﬁeld F is isometric to a trace form. Then, J.-P. Serre (in a letter to Krüskemper) pointed out that, over a number ﬁeld, every positive form of odd rank n > 4 is isometric to a trace form. Serre’s proof is based on a result of Mestre [172] concerning deformations of polynomials leaving trace forms intact. Thus, over

Quadratic Forms

49

a number ﬁeld F, all positive forms of dimension n ≥ 4 are trace forms, and for forms of lower dimension, this was known from [34, III 3.6]. For related work on trace forms, see [36, 73–76, 202, 239]. 6.2. Witt equivalence of ﬁelds Two ﬁelds with isomorphic Witt rings are called Witt equivalent. The ﬁrst question is what kind of afﬁnity of behavior of quadratic forms can we expect over two Witt equivalent ﬁelds. The following deﬁnition describes the ideal situation where quadratic forms over two ﬁelds behave in the same way. Deﬁnition 6.3. Let K and L be ﬁelds of characteristic = 2. We say that K and L are equivalent with respect to quadratic forms, when there exists a pair of bijective maps ˙ K ˙ 2 → L/ ˙ L ˙2 t : K/

and

T : Cl(K) → Cl(L),

where Cl(K) denotes the set of equivalence classes of nonsingular quadratic forms over K, satisfying the following four conditions: (A) (B) (C) (D)

˙ K ˙ 2. T(a1 , . . . , an ) = (t(a1 ), . . . , t(an )) for all a1 , . . . , an ∈ K/ det T(q) = t(det q) for every nonsingular quadratic form q over K. DL (T(q)) = t(DK (q)) for every nonsingular quadratic form q over K. t(1) = 1 and t(−1) = −1.

In 1970, Harrison [84] proved a criterion for Witt equivalence of ﬁelds, which combined with a result of Cordes [38] yielding the following theorem. Theorem 6.4 (Harrison criterion). Let K and L be ﬁelds of characteristic = 2. The following statements are equivalent: (a) K and L are equivalent with respect to quadratic forms. ˙ K ˙ 2 → L/ ˙ L ˙ 2 such that t(−1) = −1, (b) There exists a group isomorphism t : K/ 2 ˙ ˙ and for all a, b ∈ K/K , 1 ∈ DK (a, b) ⇔ 1 ∈ DL (t(a), t(b)). (c) W(K) ∼ = W(L). (d) W(K)/I 3 (K) ∼ = W(L)/I 3 (L). For proof, see [189] or [228, p. 417]. For a characteristic 2 counterpart of Harrison’s criterion, see the paper [12]. Example 6.1. Quadratically closed ﬁelds are all Witt equivalent and their Witt ring is Z/2Z. Quadratic forms are classiﬁed by their dimension. Real closed ﬁelds are all Witt equivalent and their Witt ring is Z. Quadratic forms are classiﬁed by dimension and signature.

50

K. Szymiczek

Finite ﬁelds of characteristic different from 2 are Witt equivalent if and only if they have the same level. Here quadratic forms are classiﬁed by dimension and determinant, and there are only two (kinds of ) nonisomorphic Witt rings for ﬁnite ﬁelds of characteristic = 2. These are Z/4Z for ﬁelds with the number of elements congruent to 3 (mod 4), and the group ring Z/2Z[F˙ /F˙ 2 ] for the remaining ﬁnite ﬁelds of odd characteristic. Two completions Fp1 and Fp2 of a global ﬁeld F at primes p1 , p2 are Witt equivalent if and only if |F˙ p1 /F˙ p21 | = |F˙ p2 /F˙ p22 | and s(Fp1 ) = s(Fp2 ) (see [143, Chapter VI]). Here again quadratic forms are classiﬁed by the same set of invariants (dimension, determinant, and the Hasse invariant), but there are inﬁnitely many distinct Witt rings for this class of ﬁelds (see [143, ChapterVI]). 6.3. Fields with ﬁnite square class number From Harrison’s criterion it follows that Witt equivalence preserves the order of the square class group of the ﬁeld, and so classiﬁcation of ﬁelds up to Witt equivalence reduces to classiﬁcation of ﬁelds with a given cardinality of the group of square classes. So it is natural to begin with ﬁelds having only ﬁnitely many square classes. We have already indicated in Example 6.1 that all ﬁelds with one square class are equivalent, and all ﬁelds with two square classes split into exactly three classes with respect to Witt equivalence represented by real closed ﬁelds and ﬁnite ﬁelds. These are the only simplest results in a series of classiﬁcation theorems for ﬁelds with a ﬁnite number of square classes. Write q = q(F ) = |F˙ /F˙ 2 | for the number of square classes of a ﬁeld F , and let w(q) be the number of classes of Witt equivalent ﬁelds (of characteristic = 2) with q square classes. Then, as said earlier, w(1) = 1 and w(2) = 3. The exact values of w(q) are known only for q ≤ 32. They are shown in the following table: q w(q)

1 1

2 3

4 6

8 17

16 51

32 155

Witt groups of nonreal ﬁelds with q ≤ 8 were classiﬁed by Cordes [38]. For q = 8, he found 10 distinct possible Witt groups and for 7 of them he found ﬁelds having the prescribed Witt groups. The three remaining ﬁelds were constructed by Kula [133]. For formally real ﬁelds with q = 8, the classiﬁcation was carried out in [224] and [139]. Seven possible Witt groups were exhibited and for each of them a ﬁeld with that Witt group was constructed. These results classify Witt groups, but the method of classiﬁcation is based on determining value sets of binary forms and, by Harrison’s criterion, these determine the Witt ring structure. However, Cordes [38, Example 7.2] gave an example of two ﬁelds K and F with isomorphic Witt groups and nonisomorphic Witt rings. In his example, q(K) = 8 and q(F ) = 16. In fact, there exists a simpler example of this type with q(K) = 4 and q(F ) = 8 (see the cases (3.3) and (4.1) of Theorem 3.2 in [224]). The ﬁelds with q = 16 were classiﬁed in 1979 in the dissertation of Szczepanik [221] (published in 1985). An interesting fact revealed by that classiﬁcation is that, for ﬁelds with q = 16, there are 51 distinct Witt rings, but there are only 45 distinct

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51

Witt groups. Hence, there exist ﬁelds having the same number of square classes with isomorphic Witt groups and nonisomorphic Witt rings. These observations show that the Witt ring, and not the Witt group, is the object reﬂecting properly the behavior of quadratic forms over ﬁelds. Hence, one can maintain that generalizations of Witt groups, which do not have natural ring structure, cannot be viewed as part of quadratic form theory. The classiﬁcation of Witt rings of ﬁelds with q = 32 is due to Carson and Marshall [25]. 6.4. Witt equivalence of global ﬁelds In this section, we consider global ﬁelds of characteristic not 2. These are either number ﬁelds (ﬁnite extensions of the rationals) or function ﬁelds (ﬁnite extensions of rational function ﬁelds with a ﬁnite ﬁeld of constants of characteristic not 2). The following property of global ﬁelds turns out to be important for investigating Witt equivalence of global ﬁelds. A Hilbert-symbol equivalence (HSE, for short) between two global ﬁelds K and L is a pair of maps ˙ K ˙ 2 → L/ ˙ L ˙ 2, t : K/

T : K → L ,

where t is an isomorphism of the square class groups and T is a bijective map between the sets of all primes of K and L, preserving quadratic Hilbert symbols2 in the sense that (a, b)p = (ta, tb)T p

˙ K ˙ 2, ∀ a, b ∈ K/

∀ p ∈ K .

Using standard results such as the Hasse principle for quadratic forms over global ﬁelds and the Harrison criterion for Witt equivalence of ﬁelds, one shows that HSE implies Witt equivalence of global ﬁelds. Thus, HSE is a set of local conditions for Witt equivalence of global ﬁelds. Actually, the two equivalences coincide, but the proof of the converse is not so straightforward (see [189, 226, 229] for three different proofs). The main result on Witt equivalence of global ﬁelds has the form of a local–global principle. Theorem 6.5 ([189]). Two global ﬁelds are Witt equivalent if and only if their primes can be paired so that corresponding completions are Witt equivalent. Since Witt equivalence of the completions is well understood (see Example 6.1), the local–global principle is an efﬁcient tool for checking Witt equivalence of global ﬁelds. The local behavior implies that global function ﬁelds split into two Witt equivalence classes (depending on the level) and these are disjoint from Witt equivalence classes of number ﬁelds. Two other simple consequences of the 2 For a deﬁnition of the quadratic Hilbert symbol see [143, p. 159].

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local–global principle, not obvious otherwise, are the following. First, the rational ﬁeld Q forms a singleton class, being not Witt equivalent to any other number ﬁeld, and second, the Witt equivalence of number ﬁelds preserves the ﬁeld degrees over Q. More generally, as a consequence of the local–global principle, the following is the complete set of invariants of Witt equivalence for a number ﬁeld F : n, r, s, g ; (ni , si ), i = 1, . . . , g , where n is the ﬁeld degree over Q, r is the number of inﬁnite real primes, s is the level of the ﬁeld (assuming the value zero when F is formally real), g is the number of dyadic primes, and ni and si are local dyadic degrees and the corresponding local dyadic levels, respectively (see [24]). Since n = n1 + · · · + ng , and with ﬁxed values of local degrees the other invariants can assume only ﬁnitely many values, it follows that given the degree n, there are only ﬁnitely many Witt equivalent classes of number ﬁelds of degree n. The number w(n) of Witt equivalence classes of number ﬁelds of degree n is known for n ≤ 11. For example, w(2) = 7, w(3) = 8, w(4) = 29, and w(9) = 365 (see [227]). Remark 6.6. The Witt ring of a global ﬁeld is a much more sensitive measure than the additive Witt group. One can prove that two formally real algebraic number ﬁelds have isomorphic Witt groups if and only if they have the same number of orderings. Two nonreal algebraic number ﬁelds have the same Witt group if and only if they have the same level. For quadratic ﬁelds, this gives only four Witt groups (while there are seven distinct Witt rings), and for cubic ﬁelds, we get only two Witt groups (and eight Witt rings). Another striking difference is that number ﬁelds with isomorphic Witt groups can have distinct degrees over Q. As a result of good understanding of the Witt ring of global ﬁeld, Czogała and Sładek [55] gave a presentation for the Witt ring W(F ) of a global ﬁeld of characteristic not 2 in terms of square classes, Hilbert symbols and the reduced Witt ring. This had been done earlier for Q by DeMeyer and Harrison [58, p. 9]. Hilbert-symbol equivalences come in two types: tame and wild. The equivalence (t, T ) is said to be tame at the ﬁnite prime p if ord p (a) ≡ ord T p (ta)

(mod 2),

˙ K ˙ 2 ; otherwise (t, T ) is wild at p. The equivalence (t, T ) for all square classes a ∈ K/ is said to be tame if it is tame at every ﬁnite prime of K. A main interest in tame equivalence comes from the fact that it induces the following commutative diagram with exact rows ∂K

0 → W(O ⏐ K) → ⏐

W(K) ⏐ ⏐ ϕ

−→

0 → W(OL ) →

W(L)

−→

∂L

p W(O ⏐ K /p)

⏐ ϕ

p W(OL /T p)

2 → 1 → C(K)/C(K) ⏐ ⏐

→

C(L)/C(L)2 → 1

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Here, OK is the ring of integers in K, W(OK ) is the Witt ring of OK (deﬁned in Section 9 below), and C(K) is the ideal class group of K. In this diagram, the ﬁrst three vertical arrows are ring isomorphisms and the fourth is a group isomorphism. It follows that tame equivalence preserves the integral Witt rings and also preserves the 2-ranks of ideal class groups. Tame equivalence was introduced in [189] and studied in [35, 47, 48, 130, 230]. In [231], tameness is studied in the context of semigroups with divisor theory. Hilbert-symbol equivalence has been generalized to the context of higher degree Hilbert symbols. For this, see the papers [49, 51–53, 218]. Another generalization of HSE was studied by Koprowski in [129]. In algebraic function ﬁelds of one variable over a ﬁxed real closed ﬁeld k, Hilbert symbols are replaced with quaternion algebras. It is proved that there are only two Witt rings for this class of ﬁelds so that all formally real function ﬁelds over k are Witt equivalent. It is established that two formally real function ﬁelds are tamely equivalent if and only if the associated algebraic curves have the same number of semialgebraically connected components. The genus of a function ﬁeld turns out not to be an invariant of tame equivalence. And ﬁnally, there is one special case: for R, the ﬁeld of real numbers, the rational function ﬁeld k(X) is tamely equivalent to R(X) if and only if k is isomorphic to R. In 1985, Baeza and Moresi showed that any two global ﬁelds of characteristic 2 are Witt equivalent (see [12]). It is not difﬁcult to see that a global ﬁeld of characteristic 2 is never Witt equivalent to a global ﬁeld of characteristic different from 2. 7. Function ﬁelds 7.1. Selected applications of function ﬁelds techniques A nonsingular quadratic form q of dimension at least 2 over a ﬁeld F , viewed as a polynomial over F , is irreducible over F unless q is a hyperbolic plane. Hence, assuming that dim q ≥ 2 and q is not a hyperbolic plane, we can consider the integral domain F [X]/q(X) called the afﬁne algebra of the quadric q = 0 and its ﬁeld of fractions F [q] called the function ﬁeld of q. It is obvious that q becomes isotropic over F [q], but it is a nontrivial question as to what other quadratic forms g over F become isotropic over F [q]. If g becomes isotropic over F [q], one writes g > q, and if g becomes hyperbolic over F [q], one writes g >> q. The starting point of the theory is the following result. Theorem 7.1. If g >> q and 1 ∈ DF (q), then q(X) · g ∼ = g over the rational function ﬁeld F(X). Moreover, if g is anisotropic, then dim g ≥ dim q. From this result, the Arason–Pﬁster Hauptsatz follows without much difﬁculty by an inductive argument (see [143, p. 354]). Since an isotropic Pﬁster form q is hyperbolic, it becomes hyperbolic over F [q]. Conversely, from Theorem 7.1 and

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the multiplicative properties of Pﬁster forms it follows that, if q >> q, then q is either a scalar multiple of a Pﬁster form or is itself a hyperbolic form. According to Lam [143, p. 348], the following three results are amongst the strongest and most interesting results obtained on function ﬁelds in the decade. Theorem 7.2 (Hoffmann [92]). Let q and g be quadratic forms over F , and let g be anisotropic. If the dimensions of q and g are separated by a power of two, that is dim g ≤ 2n < dim q for some n ≥ 1, then g remains anisotropic over F [q]. The ﬁrst Witt index i1 (q) of an anisotropic quadratic form q over F is by deﬁnition the (usual) Witt index of the form q over its function ﬁeld F [q]. Clearly, i1 (q) ≥ 1, and the precise determination of all possible values of the ﬁrst Witt index is given in the following theorem. Theorem 7.3 (Karpenko [110]). Let q be an anisotropic quadratic form, and let dim q − 1 = 2n1 + 2n2 + · · · + 2nr be the dyadic expansion of dim q − 1, n1 < n2 < · · · < nr . Then, for the ﬁrst Witt index i1 (q), we necessarily have i1 (q) − 1 = 2n1 + 2n2 + · · · + 2ns for some nonnegative s < r. For an anisotropic form q the integer dimes q = dim q − i1 (q) + 1 is said to be the essential dimension of q. Theorem 7.4 (Karpenko and Merkurjev [113]). For anisotropic forms g and q over F , if g > q, then dimes g ≥ dimes q and equality holds iff q > g. As an application of their result, Karpenko and Merkurjev proved the following extension of the Arason–Pﬁster Hauptsatz. Theorem 7.5. For a positive-dimensional anisotropic quadratic form q over a ﬁeld F , if q ∈ I n (F ) and dim q > 2n , then dim q ≥ 2n + 2n−1 . While Karpenko and Merkurjev’s arguments work in any characteristic = 2, Theorem 7.5 was ﬁrst proved for ﬁelds F of characteristic 0 by Vishik [234]. There is an even more penetrating result solving completely the problem of determining dimensions of anisotropic forms q in I n (F ). Theorem 7.6 (Vishik gap theorem). Any anisotropic form q in I n (F ) has dimension equal to one of the following numbers: 2n+1 − 2r+1 ,

0 ≤ r ≤ n or 2n+1 + 2k,

k ≥ 0.

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It is also possible to realize all these dimensions by anisotropic forms in I n (F ) over a suitable ﬁeld F. For a proof of Vishik’s gap theorem see Karpenko [111]. Knebusch [119] introduced generic splitting towers of quadratic forms, which is essentially the following construction leading to higher Witt indices ij (q) by iterating the process of forming the ﬁrst Witt index i1 (q) (for ﬁelds of arbitrary characteristic see [124]). So, we put F0 = F, q0 = q, and let F1 = F [q] be the function ﬁeld of q. We have deﬁned i1 (q) to be the Witt index of q over F1 , and if, for j > 0, qj−1 is the anisotropic part of q over Fj−1 , we set ij (q) to be the Witt index of qj−1 over Fj = Fj−1 [qj−1 ] if dim qj−1 ≥ 2. The smallest h such that dim qh ≤ 1 is called the height of q and i1 (q), . . . , ih (q) is the sequence of higher Witt indices of q. The question of possible values for these higher Witt indices remains open. In [112], Karpenko found relations among the 2-adic orders v2 (ij (q)) of the higher Witt indices. He proved that max v2 (ij (q)) ≥ v2 (im (q)) ≥ min v2 (ij (q)) − 1,

m<j≤h

m<j≤h

where the ﬁrst inequality requires that im (q) + 2 hj=m+1 ij (q) be not a power of two. See also [98] for the lower bound of the heights of all forms of a given dimension. As a ﬁnal example, we discuss characterization of I n (F ) via Knebusch’s degree. For an even-dimensional form q of height h, the form qh−1 above has itself height 1 and so it is similar to a Pﬁster form by a theorem of Wadsworth [237]. If qh−1 is similar to an n-fold Pﬁster form, n = deg q is said to be the Knebusch’s degree of the form q. Observe that, when q is an n-fold Pﬁster form, then its height is 0 and hence its degree equals n. Now, following Knebusch [119], we set Jn (F ) = {q ∈ I(F ) : deg q ≥ n} . Knebusch [119, Theorem 6.4] proved that Jn (F ) is an ideal of the Witt ring W(F ). A ﬁrst consequence of this is that I n (F ) ⊆ Jn (F ). To see this, observe that for an anisotropic form q we have dim q ≥ 2n , where n = deg q. Since all n-fold Pﬁster forms have degree n, they certainly belong to Jn (F ), and so, since Jn (F ) is an ideal of the Witt ring W(F ), we have I n (F ) ⊆ Jn (F ). A second consequence is the Arason–Pﬁster Hauptsatz. For, if I n (F ) ⊆ Jn (F ), then each anisotropic form in I n (F ) has degree at least n, and hence the dimension at least 2n , which is the Arason–Pﬁster Hauptsatz. A natural and important question to ask was whether I n (F ) = Jn (F ) holds for all n ≥ 1. This outstanding problem was solved in the afﬁrmative in Orlov–Vishik– Voevodsky’s paper [182, Theorem 4.3]. There are several surveys and texts on quadratic forms emphasizing function ﬁelds and their applications in quadratic form theory. First, Lam [143] gives an introduction to the subject and an overview of new results in the area. The Lens Conference volume [100] includes an introduction to motives of quadrics by Vishik, with applications to the splitting patterns of quadratic forms; papers by Izhboldin and Karpenko on Chow

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groups of quadrics with application to the construction of ﬁelds with u-invariant 9; and a paper by Kahn presenting a general framework for the computation of the unramiﬁed cohomology groups of quadrics and other varieties. A forthcoming book [65] by Elman, Karpenko and Merkurjev is a research monograph written from the viewpoint of algebraic geometry and presents quadratic form theory in full generality. Part I of the book was available before publication in preprint form. The publisher’s announcement includes the following information. “Part I includes the classical algebraic theory of quadratic and bilinear forms and answers many questions that have been raised in the early stages of the development of the theory. Assuming only a basic course in algebraic geometry, Part II presents the necessary additional topics from algebraic geometry including the theory of Chow groups, Chow motives, and Steenrod operations. These topics are used in Part III to develop a modern geometric theory of quadratic forms.”

7.2. u-invariant For nonreal ﬁeld F, Kaplansky deﬁned u(F ) to be the maximal dimension of universal quadratic form (which is the same as the maximal dimension of an anisotropic quadratic form) over F . Here, universal means representing all nonzero ﬁeld elements. Hence, u(F ) = 1 iff F is quadratically closed, u(F ) = 2 for a ﬁnite ﬁeld F , u(F ) = 4 for any local ﬁeld F , and hence (by Hasse Principle) u(F ) = 4 for any nonreal global ﬁeld F . If F is an algebraic function ﬁeld of transcendence degree n over the complex ﬁeld C, then u(F ) ≤ 2n and u(F ) = 2n when F is ﬁnitely generated over C. This is a special case of a more general result proved in 1936 by Tsen [233]. However, using the theory of Pﬁster forms one easily proves that always u(F ) = 3, 5, 7. In 1953, Kaplansky [108] conjectured that u(F ) is always a power of two (or ∞). He observed that, for a ﬁeld F complete with respect to a discrete rank one valuation with residue class ﬁeld K of characteristic different from two, u(F ) = 2u(K). This produces examples of ﬁelds with u-invariant equal to any given power of 2 (for instance, the iterated power series ﬁeld C((t1 )) · · · ((tn )) has u = 2n ). See also the Elman and Lam papers [66, 67]. Kaplansky’s conjecture was proved for various classes of ﬁelds, but in 1988, much to anybody’s surprise, the ﬁrst examples of nonreal ﬁelds with u-invariant equal to 6 were constructed by Merkurjev [169]. Actually, any ﬁeld K is contained in a ﬁeld F with u(F ) = 6. Merkurjev’s construction uses function ﬁeld techniques for six-dimensional quadratic forms of determinant arise naturally as “Albert −1. These a,b c,d forms” of biquaternion algebras. If B = F and C = F are quaternion algebras over F , then the Albert form of their tensor product A = B ⊗F C is the form qA = (−a, −b, ab, c, d, −cd). The crucial property of qA is that, A is a division algebra iff qA is anisotropic over F (see [127, Chapter IV, Section 16] for a comprehensive discussion of Albert forms). The main ingredient of Merkurjev’s construction is the following theorem.

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Theorem 7.7 (Merkurjev). If the Albert form qA is anisotropic over F , then for any nonsingular form g over F of dimension ≥ 7, the form qA remains anisotropic over the function ﬁeld F [g]. For a comprehensive survey and details of proofs, see Lam [144] and also [143, pp. 484–495]. Further, in 1991, Merkurjev [170] proved that for any even number 2n there exists a ﬁeld (of cohomological dimension two) with u-invariant 2n. This construction led to the study of index-reduction formulas for division algebras and algebraic groups. In case of ﬁelds of characteristic 2, ﬁelds with any even u-invariant were constructed by Mammone, Tignol, and Wadsworth [160] using the ideas of Merkurjev’s construction of ﬁelds with u = 6. The next challenge was to decide whether there exist ﬁelds with odd u-invariant. Izhboldin [99] constructed the ﬁrst examples of ﬁelds with u = 9. It is now known that u(F ) can assume any value of the form 2r + 1 for r ≥ 3. This was proved by Vishik [235] using a new method which also allows constructing ﬁelds with even u-invariant without using the Merkurjev index reduction formula. Anatural problem in the study of the u-invariant is to ﬁnd estimates for u under ﬁnite ﬁeld extensions. The best general result in this direction is the following theorem. Theorem 7.8 (Leep [145]). For a nonreal ﬁeld F and any ﬁeld extension K/F of degree n = [K : F ] < ∞, we have u(K) ≤ 12 (n + 1)u(F ). The proof depends on a generalization of the u-invariant to the context of systems of quadratic forms. See [147, 177]. The u-invariant considered so far is an invariant of nonreal ﬁelds. Already Kaplansky in 1953 stated that “the formally real case probably has a parallel theory, which may be worth separate study.” For a formally real ﬁeld F , there are at least three distinct generalizations of the u-invariant. We shall conﬁne attention to the one introduced by Elman and Lam in [66]. The u-invariant of F is the supremum of the dimensions of anisotropic forms over F having signature zero with respect to every ordering of F . By the Pﬁster local–global principle (Theorem 3.3), u(F ) is the supremum of dimensions of anisotropic torsion forms over F . Torsion forms are necessarily even-dimensional, so u(F ), if ﬁnite, is an even integer. For power series ﬁelds, the equality u(F((t))) = 2u(F ) still holds; hence, there are formally real ﬁelds with the u-invariant 2n for any integer n. But also here Kaplansky’s conjecture fails to hold. Modifying Merkurjev’s construction Lam [144, Theorem 5.2] shows that there exist formally real ﬁelds with u-invariant 6. Then, in a (handwritten) manuscript, “Some consequences of Merkurjev’s work on function ﬁelds” (1989), Lam adopted the arguments of the preprint version of Merkurjev’s paper [170] to the formally real ﬁeld context and proved that there exist formally real ﬁelds with u = 2n, any given even integer. Similar results were published a little later by Hornix [96]. A survey of results on isotropy questions over function ﬁelds of quadrics is available (see Hoffmann [94]). It presents a uniform approach to constructing ﬁelds with prescribed invariants (level, Pythagoras number, u-invariant, including u = 9).

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8. Sums of squares 8.1. Level Recall that for a ring A, the level s(A) is the smallest natural number n such that −1 ∈ A is a sum of n squares in A or ∞ if −1 is not a sum of squares in A. We have mentioned already in Section 2 that the level of a nonformally real ﬁeld is always a power of two. This was proved by Pﬁster as a simple consequence of the fact that for a Pﬁster form q, the value set DF (q) is a group under multiplication. In particular, nonzero sums of 2n squares of elements of F form a group under multiplication. Now, assume that F is a nonreal ﬁeld with s = s(F ) and let 2n ≤ s < 2n+1 . Then −1 = A + B, where A is a sum of 2n and B a sum of less than 2n squares in F . By the group property, it follows that −1 = (1 + B)/A is a sum of 2n squares in F , and so s = 2n . This proves the ﬁrst part of the following theorem. Theorem 8.1 (Pﬁster level theorem [190]). For any nonreal ﬁeld F , its level s(F ) is a power of 2. For any s = 2n , there exists a nonreal ﬁeld F with s(F ) = s. 2 over R, where The function ﬁeld R[q] of the quadratic form q = X12 + · · · + Xm n+1 n ≤ m − 1 < 2 , can be shown to have level 2 . Pﬁster’s level theorem, together with the calculations of the u-invariant and Pythagoras number prior to Merkurjev’s breakthrough in 1989, gave the false impression that all sensible quadratic form invariants of ﬁelds should be powers of two.3 In particular, it would be natural to expect that the level of integral domains also assumes only some restricted values. However, there are no restrictions at all, as follows from the following elegant theorem.

2n

Theorem 8.2 (Dai–Lam–Peng [56]). s(R[X1 , . . . , Xn ]/(1 + X12 + · · · + Xn2 )) = n. The method of proof is somewhat astonishing: it is based on the topological Borsuk– Ulam antipodal point theorem. Dai and Lam [57] studied in depth the interrelations between levels in algebra and topology. In topology, the level s(X) of a topological space X endowed with a ﬁx-point-free involution i is the smallest number n with the property that there exists an equivariant map from (X, i) to the sphere (S n−1 , −), where − denotes the antipodal involution on the sphere (or s(X) = ∞ if such a map does not exist). Then, for instance, the Borsuk–Ulam theorem states that s(S n−1 , −) = n. Dai and Lam prove a general “Level Theorem” associating with each space (X, i) a commutative R−algebra AX such that the levels coincide: s(X) = s(AX ). Theorem 8.2 is then obtained by taking X = S n−1 . For proofs, see also [143, pp. 505–514] and [198, pp. 46–50]. 3 At an Oberwolfach conference on quadratic forms sitting at breakfast, I mentioned in the presence of J.-P. Serre that it is a bit strange that the organizers invited only 29 persons for the conference since in quadratic form theory all good invariants are powers of two. Serre replied: 16 would be enough.

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Lewis [150] constructed examples of quaternion division algebras whose levels take any prescribed value 2k or 2k + 1. Whether the level of a division quaternion algebra can take other values was an open question. Recently, Hoffmann [95] proved that there are inﬁnitely many values not of the form 2k or 2k + 1 that occur as levels of quaternion algebras. This uses function ﬁeld techniques and a result of Karpenko and Merkurjev [113]. For a survey on levels of ﬁelds and quaternion algebras, see Lewis [156] and the list of references there. For the relation between the commutativity of a division algebra and representability of 0 as a sum of squares, see [148].

8.2. Composition A composition formula of size [r, s, n] is a sum of squares formula of the type (x12 + x22 + · · · + xr2 ) · (y12 + y22 + · · · + ys2 ) = z21 + z22 + · · · + z2n , where X = (x1 , x2 , . . . , xr ) and Y = (y1 , y2 , . . . , ys ) are systems of indeterminates and each zk = zk (X, Y) is a bilinear form in X and Y . A triple [r, s, n] is admissible over a ﬁeld F if a composition formula of size [r, s, n] exists with the bilinear forms zk in X and Y having coefﬁcients in F . Formulas of size [n, n, n] are known to exist for n = 1, 2, 4, 8, and they are related to multiplication in the ﬁeld of complex numbers, and in quaternion and octonion algebras. Hurwitz proved in 1898 that [n, n, n] formulas exist over C only for n = 1, 2, 4, 8. Actually, Hurwitz’s arguments work over any ﬁeld of characteristic = 2. Another classical result is the Hurwitz–Radon theorem for the size [r, n, n]. For this to be admissible over R or C, it is necessary and sufﬁcient that r ≤ ρ(n), where ρ(n) is the “Hurwitz–Radon function” deﬁned as follows: if n = 24a+b n0 , where n0 is odd and 0 ≤ b ≤ 3, then ρ(n) = 8a + 2b . Again, Hurwitz’s ideas can be generalized to any ﬁeld of characteristic = 2. In 1940, using topological methods, Hopf proved the following result. Theorem 8.3. If [r, s, n] is admissible over R, then the binomial coefﬁcient even whenever n − s < k < r.

n k

is

The parity condition on binomial coefﬁcients in the above theorem is said to be the hopf condition. It has been known that for any formally real ﬁeld F , and even for all ﬁelds of characteristic 0, if [r, s, n] is admissible over F , then the Hopf condition holds. However, generalizing this to ﬁelds of prime characteristic was an unsolved problem. As it turned out recently, extending Hopf’s theorem to ﬁelds of prime characteristic required the use of motivic cohomology. Dugger and Isaksen [64] proved in 2007 that the Hopf conditions are necessary for admissibility of [r, s, n] over an arbitrary ﬁeld F of characteristic = 2. A comprehensive presentation of all aspects of the composition of quadratic forms is contained in the book Shapiro [209]. See also [127, Chapter VIII].

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8.3. Pythagoras number The Pythagoras number of a ring R, denoted P(R), is the smallest positive integer n such that every sum of squares in R can be written as a sum of ≤n squares. The Pythagoras number of a nonreal ﬁeld F with level s assumes one of the following two values: s or s + 1. Indeed, any element of the ﬁeld can be written as a difference of two squares x2 + (−1)y2 and −1 is a sum of s squares. Moreover, using standard arguments, one can show that for any s = 2n there are ﬁelds of level s and Pythagoras number s, s + 1, respectively (see [143, p. 396]). The formally real ﬁelds have a much more difﬁcult and interesting theory of Pythagoras numbers. Prestel [201] showed that for any number m, there exist uniquely ordered formally real subﬁelds F and K of R with P(F ) = 2m and P(K) = 2m + 1. For a long time, it remained an open problem whether the Pythagoras number of a formally real ﬁeld, if ﬁnite, must be of the form 2m or 2m + 1. A complete solution to this problem came as a great surprise not so because of the author but rather because of the answer. Theorem 8.4 (Hoffmann [93]). For any n ≥ 1, there exists a formally real ﬁeld F with P(F ) = n. In fact, a more precise result was obtained, namely a ﬁeld exists which is uniquely ordered, has P(F ) = n and has any prescribed even value of the u−invariant. The method of proof relies on function ﬁeld technique used by Merkurjev to prove the existence of ﬁelds with u = 6. Lam [143, pp. 495–499] explains the approach in the example of construction of ﬁelds with Pythagoras numbers 6 and 7. Artin [8] proved in 1927 that every positive semideﬁnite rational function in K = R(X1 , . . . , Xn ) is a sum of squares of rational functions. His approach does not say, however, whether the Pythagoras number P(K) is ﬁnite. This was clariﬁed 40 years later by Pﬁster. Theorem 8.5 (Pﬁster [192]). P(R(X1 , . . . , Xn )) ≤ 2n . The three main ingredients of Pﬁster’s proof are the Artin result, the theorem of Tsen, and Pﬁster’s theory of multiplicative (Pﬁster) forms. While this is an important quantitative result, yet all is not well. We know the exact value of P(R(X1 , . . . , Xn )) only for n = 1 and n = 2. Here, P(R(X1 )) = 2 is elementary algebra, and P(R(X1 , X2 )) = 4 is a very difﬁcult result proved by Cassels, Ellison and Pﬁster [27] in 1971 by using the theory of elliptic curves over R(X). On the other hand, for the polynomial rings over R and Z, the situation is clariﬁed by the following theorem. Theorem 8.6 (Choi, Dai, Lam, and Reznick [28]). P(R[X1 , . . . , Xn ]) = ∞ for n ≥ 2 and P(Z[X1 , . . . , Xn ]) = ∞ for n ≥ 1. Another natural question is to ﬁnd the Pythagoras numbers P(Q(X1 , . . . , Xn )) and P(Q[X1 , . . . , Xn ]). Landau proved in 1906 that P(Q[X1 ]) ≤ 8 and Pourchet [200] in 1971 established P(Q[X1 ]) = 5. According to a result of Cassels [26], for an

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arbitrary ﬁeld F , one has P(F [X1 ]) = P(F(X1 )). Hence, P(Q(X1 )) = 5. For n = 2 and n = 3, it was proved by Colliot-Thélène and Jannsen [32] that P(Q(X1 , X2 )) ≤ 8

and P(Q(X1 , X2 , X3 )) ≤ 16.

A conditional result in [32], modulo Milnor’s conjecture and a conjecture of Kato, asserts that P(Q(X1 , . . . , Xn )) ≤ 2n+1 for n ≥ 2. For further details, see [199] and [77]. 9. Witt rings of rings 9.1. Witt rings of commutative rings Knebusch [115] generalized in 1970 the construction of the Witt ring W(R) of a ﬁeld R to cover the case when R is a commutative ring. Instead of nonsingular symmetric bilinear forms on ﬁnite dimensional vector spaces, one considers ﬁnitely generated projective modules over R equipped with nonsingular symmetric bilinear forms, called inner product spaces. For two inner product spaces (S, α) and (T, β) their orthogonal direct sum, written (S, α) ⊥ (T, β) or S ⊥ T , is deﬁned to be the direct sum of the modules S and T with bilinear form γ, deﬁned by the equation γ(s1 ⊕ t1 , s2 ⊕ t2 ) = α(s1 , s2 ) + β(t1 , t2 ). For inner product spaces S and T , S ⊥ T is also an inner product space. For an inner product space (S, α) and for a submodule N of S, consider the orthogonal complement N ⊥ of N in S, N ⊥ := {s ∈ S : α(s, N) = 0} . The inner product space (S, α) is said to be metabolic if there exist submodules M and N of S such that, S =M⊕N

and

N = N ⊥.

Two inner product spaces M and N are said to be similar, or Witt equivalent, written as M ∼ N, if there exist metabolic spaces S and T so that M ⊥ S is isometric to N ⊥ T . Similarity is an equivalence relation and we write W(R) for the collection of all similarity classes (Witt classes) of inner product spaces over the commutative ring R. The class containing the space (M, β) is written as M, β or simply M. We deﬁne two operations in W(R), the addition and the multiplication of classes by setting M + N = M ⊥ N and

M · N = M ⊗ N.

These operations do not depend on the choice of representatives of the similarity classes, and the sum and the product of similarity classes of two inner product spaces are the similarity classes of inner product spaces. Thus, we have well-deﬁned addition and multiplication operations in W(R). The collection W(R) of all Witt classes of inner

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product spaces over R forms a commutative ring, using the orthogonal sum as addition operation and the tensor product as multiplication operation. When R is a ﬁeld, this construction is identical with the original construction of Witt presented in Section 3. Details of the construction and further material may be found in [10, 163, 174, 204]. Any ring homomorphism R → R induces a homomorphism W(R) → W(R ) of Witt rings. For an integral domain R and its ﬁeld of fractions F , consider the ring homomorphism ϕ : W(R) → W(F ) induced by the inclusion R → F . If R is a Dedekind domain it is known that ker ϕ is zero. This was ﬁrst proved by M. Knebusch ([115, Satz 11.1.1]). Also in some other cases, this homomorphism is injective. The most general situation known where this happens is the case when R is a regular domain of dimension ≤ 3 (theorem of Pardon (1984) and Ojanguren (1982), see [126, p. 472]). However, ϕ is no longer injective when the dimension of R is four. A counterexample can be found in [128]. A more general case was considered by Craven, Rosenberg, and Ware [41]. They proved that when R is a regular Noetherian domain of an arbitrary Krull dimension, then ker ϕ is a nilideal. Here, the regularity condition is not necessary since, as shown in [30], the kernel of ϕ is a nilideal also for nonmaximal orders in global ﬁelds, and these are not regular rings. On the other hand, for each order Z[fi], f > 1, of the Gaussian ﬁeld Q(i), the natural ring homomorphism W(Z[fi]) → W(Q(i)) is not injective, and there are other results of that type [31]. They conﬁrm in part the conjecture that for an algebraic number ﬁeld F and its order O the natural ring homomorphism W(O) → W(F ) is injective if and only if O is the maximal order of F . All the results on nilpotency of the kernel of natural homomorphisms of Witt rings depend on the following local–global principle proved by Dress [62]. For a ring R, and a prime ideal p of R, we write Rp for the localization of R at p. For a commutative ring R and any inner product R-space E, the class E ∈ W(R) is nilpotent if and only if ERp ∈ W(Rp ) is nilpotent for all maximal ideals p in R. From this result, facts about prime ideal structure in Witt rings over ﬁelds can be extended to commutative rings (see, [163]). Knebusch, Rosenberg, and Ware in [122, 123] generalized most of the results in Section 3 to the case of Witt rings of semilocal rings (for this, see also [116]), and of Dedekind rings. Actually, they consider semilocal and Dedekind rings with involutions and work with more general Witt rings described in [121]. Their work culminated in two papers by Knebusch [117, 118], where ﬁnally the notion of a real closure of a commutative ring was discovered and a ring-theoretical counterpart of the Artin–Schreier theory of formally real ﬁelds with its relation to Witt rings of ﬁelds was found. For a self-contained treatment of quadratic and bilinear forms over semilocal rings and of Witt rings of semilocal rings, see Baeza [10] and Marshall [163].

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9.2. Witt rings of Dedekind domains Let F be the ﬁeld of fractions of a Dedekind domain O. As mentioned in the preceding section, Knebusch [115] proved that the natural ring homomorphism i

W(O) −→ W(F ) is injective. Milnor extended the result in the following way. With p running over the set of all nonzero prime ideals of O, we have the following exact sequence for the Witt groups W(O), W(F ): i ∂ W(O/p), (9.1) 0 → W(O) −→ W(F ) −→ p

where ∂ = ∂O is the total residue homomorphism. This is said to be the Knebusch– Milnor exact sequence. The background, deﬁnitions of the maps involved, and proofs can be found in [174, Ch. IV]. The sequence (9.1) can be extended to the right in some important special cases. First of all, when O is the ring of integers of a number ﬁeld F and C = C(F ) is the ideal class group, Milnor proved that with a suitable choice of the homomorphism λ the sequence i ∂ λ 0 → W(O) −→ W(F ) −→ W(O/p) −→ C/C2 → 0 (9.2) p

is exact. Hence, the cokernel of the total residue homomorphism ∂ is the ﬁnite elementary Abelian 2-group C/C2 . For a proof, see [174, pp. 93–94] and [204, p. 227]. When O = k[X] is the ring of polynomials in one indeterminate over a ﬁeld k and F = k(X) is the ﬁeld of rational functions over k, the cokernel is known to be the zero group (Milnor’s theorem, see [204, p. 211]). When O is the ring of polynomial functions on the circle O = R[X, Y ]/(X2 + Y 2 − 1), then coker ∂ = Z is the inﬁnite cyclic group ([174, p. 94]). Pﬁster [197] studied√the Knebusch–Milnor sequence in the case of the quadratic 2 extension √ O = k[X, aX + b ] of the polynomial ring k[X] such that for F = 2 k(X, aX + b ), the ﬁeld of fractions, F/k is a function ﬁeld of genus 0 but is not a rational function ﬁeld. Pﬁster [197, Theorem 5] proved that in that case coker ∂ ∼ = ann1, −a, −b, ab/ ann1, −a, where the annihilators are taken in the Witt ring W(k). Using this result, one can construct a Dedekind domain O with the cokernel of ∂O containing torsion elements of any given two-power order (see [232] for details). The description of the cokernels of the total residue homomorphisms ∂O for all Dedekind rings O seems to be out of reach. A more realistic open problem is to characterize the class of Abelian groups, which occur as cokernels of total residue homomorphisms ∂O for all Dedekind rings O. Very little is known about Witt equivalence for general Dedekind domains. The structure of Witt rings of Hasse domains in global ﬁelds is determined in [50], and

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also, sufﬁcient conditions for Witt equivalence of Hasse domains are given. In [54], an analog of the Theorem 6.5 was proved for Witt equivalence of semilocal subrings of a Dedekind domain. 9.3. Witt rings of skew ﬁelds and algebras For a skew ﬁeld A, Craven [46] deﬁned the Witt ring W(A) as the abstract Witt ring (see Section 10.1 below) W(A) = Z[S(A)]/J, where S(A) is the subgroup of the multiplicative group A× generated by the set of 2 all squares A× . Here, J is the ideal generated by the set {[1] + [a] − [1 + a] − [a(1 + a)] : a ∈ S(A), a = 0, −1} ∪ {[1] + [−1]}. As in the commutative case, for an ordered skew ﬁeld A there is a bijective correspondence between the set of minimal prime ideals of the ring W(A) and the set X(A) of all orderings of A. The set X(A) can be given the induced Zariski topology from the prime spectrum and this makes X(A) into a Boolean space. Craven [45] proved that every Boolean space can be obtained as the space of orderings of a noncommutative skew ﬁeld. He also proved that every ﬁeld F can be embedded in a noncommutative skew ﬁeld A such that the induced ring homomorphism W(F ) → W(A) is an isomorphism. However in [46], an example is produced of a skew ﬁeld D, for which, given any commutative subﬁeld F → D, the induced ring homomorphism W(F ) → W(D) has nontrivial kernel. Another interesting question is whether for a given noncommutative skew ﬁeld A, there is a ﬁeld F such that W(A) ∼ = W(F ). Sładek proved that this is so when (a) A is a quaternion algebra over a Pythagorean ﬁeld or over a local ﬁeld ([213]) or (b) A is a division algebra over a global ﬁeld and has odd dimension ([216]). When the dimension of A is even, W(A) is a homomorphic image of the Witt ring of a ﬁeld ([217]). Sładek [215] proved a noncommutative counterpart of Springer’s theorem on the structure of Witt groups (and rings) of ﬁelds complete with respect to a discrete valuation. As a consequence, he showed that there is a noncommutative ﬁeld A complete with respect to a discrete valuation and such that the Witt ring W(A) is not isomorphic to the Witt ring of any ﬁeld ([214]). For Witt rings of central simple algebras, see Lewis and Tignol [157]. For orderings of higher level and reduced Witt rings of skew ﬁelds, see Craven [46]. 9.4. More general Witt rings and groups Knebusch deﬁned in [120] the Witt rings of arbitrary schemes. Balmer [13, 14] introduced Witt groups of triangulated categories with duality. These generalize Knebusch’s Witt groups of schemes. A discussion of the interrelations amongst various generalizations of the concept of Witt group can be found in Balmer [15].

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10. Abstract Witt rings 10.1. Elementary type conjecture We have mentioned in Section 3 that the Witt ring of a ﬁeld F has a presentation by generators and relations W(F ) = Z[G]/J, where G = F˙ /F˙ 2 is the square class group of F and J is an ideal of the group ring Z[G]. It is an important property of the Witt ring that the torsion subgroup of the additive group of the ring is 2-primary. These observations led Knebusch, Rosenberg, and Ware [121] to the following deﬁnition. An abstract Witt ring is a ring R with a presentation R = Z[G]/J, where G is an elementary 2-group and the subgroup of torsion elements in R is 2-primary. It turns out that for such abstract Witt rings one can prove most of the structure results, which Pﬁster proved for the Witt rings of ﬁelds. Abstract Witt rings form a category where a morphism h : Z[G]/J → Z[G1 ]/J1 between two objects is a ring homomorphism sending each coset g + J for g ∈ G onto a coset g1 + J1 for some g1 ∈ G1 . Denote this category as AWR. Clearly, Witt rings of ﬁelds are objects of this category. If R is an abstract Witt ring and there exists a ﬁeld F such that R and W(F ) are isomorphic, then we say that the Witt ring R is realized by the ﬁeld F . Another approach to abstract Witt rings uses linked quaternionic mappings as the basic concept of the theory (see [162, 167]). The categories of abstract Witt rings and linked quaternionic mappings are equivalent. The ﬁrst fact showing advantages of the abstract approach to Witt rings is the following observation: ﬁnite products exist in the category AWR. Indeed, given abstract Witt rings R1 , . . . , Rn , the subring R of the Cartesian product R1 × · · · × Rn consisting of elements (f1 , . . . , fn ) satisfying dim fi ≡ dim fj (mod 2) for all i, j ≤ n, is the product of R1 , . . . , Rn in the category AWR (ﬁber product over Z/2Z). There is a second operation in APW, the group extension of a Witt ring. If R is a Witt ring, then the group ring R[], where is an elementary 2-group, is also an abstract Witt ring. These two operations are said to be the elementary operations on abstract Witt rings. For a given list of basic Witt rings, say R1 , . . . , Rm , the elementary operations generate a class of Witt rings constructible from the basic rings by means of the elementary operations. For example, all Witt rings of ﬁelds with square class number q ≤ 8 can be obtained by elementary operations from the following list of basic Witt rings: Z,

Z/2Z,

Z/4Z,

L,

where L is the Witt ring of the ﬁeld Q2 of 2-adic numbers. The four basic rings cannot be constructed from any simpler Witt rings by elementary operations and they are realized by the ﬁelds R, C, F3 , Q2 , respectively. A further analysis of the situation leads to the conclusion that the list of basic Witt rings should include also the Witt rings Ln , n ≥ 1, of ﬁnite extensions of degree n of the ﬁeld Q2 of 2-adic numbers.

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If n = 2k is even, there are two nonisomorphic Witt rings denoted L2k,0 and L2k,1 corresponding to ﬁelds with level s = 1 or 2, respectively. All these Witt rings are indecomposable in terms of elementary operations. The Witt rings obtained from Z,

Z/2Z,

Z/4Z,

L2k,0 ,

L2k,1 ,

L2k−1

k≥2

by elementary operations are said to be of elementary type. With this list of basic Witt rings, we have the following elementary type conjecture: Each ﬁnitely generated abstract Witt ring is of elementary type. A restricted form of the conjecture states that each ﬁnitely generated abstract Witt ring, which is realized by a ﬁeld, is of elementary type. The classiﬁcation results for ﬁelds with respect to Witt equivalence described in Section 6.3 conﬁrm the ETC for Witt rings of ﬁelds with square class number q(F ) ≤ 32. The ETC has also been proved for torsion-free abstract Witt rings. As we know from Theorem 3.4, the Witt ring of a ﬁeld is torsion-free if and only if the ﬁeld is Pythagorean. Hence, the Witt ring of any Pythagorean ﬁeld with ﬁnite square class number can be obtained from the Witt ring of the reals W(R) = Z by means of a sequence of elementary operations. This was ﬁrst proved by Bröcker [23] and Craven [43, 44] in the ﬁeld case (for reduced Witt rings, see Section 10.2 below) and then by Marshall ([161]), and independently in the dissertation of Sładek in 1979, in the general abstract case. Elementary operations and ETC are particularly (and, perhaps, exclusively) important for Witt rings of ﬁelds (abstract Witt rings realizable by ﬁelds). A fundamental question arises whether rings obtained by elementary operations from Witt rings realizable by ﬁelds are likewise realizable by ﬁelds. In case of group extensions, a positive answer follows from results of Springer [220] on the structure of the Witt group of a nondyadic complete discretely valuated ﬁeld F . Springer’s theorem asserts that W(F ) ∼ = W(F ) ⊕ W(F ), where F is the residue class ﬁeld of F (for an analog of Springer’s theorem for ﬁelds of characteristic 0, carrying a 2-Henselian valuation with residue ﬁeld of characteristic 2, see Jacob [101, 102]). Roughly speaking, group extensions of the Witt ring W(F ) are realized by (iterated) formal power series ﬁelds over F . Realizability of products is more difﬁcult and was proved by Kula using valuation theoretic means. Theorem 10.1 (Kula [132, 134]). The product of Witt rings of ﬁelds is the Witt ring of a ﬁeld. This theorem and the validity of ETC would give a complete solution to the classiﬁcation problem of Witt rings of ﬁelds with ﬁnite groups of square classes. The complexity of ETC for the Witt ring of a ﬁeld F is measured by the square class number q(F ) = |F˙ /F˙ 2 |. As we have mentioned earlier, for q ≤ 32, ETC holds. Another measure of complexity of ETC for a ﬁeld F is the number Q = Q(F ) of quaternion algebras over F . The most difﬁcult are the cases where Q is a power of two. The case Q = 2 is already nontrivial and was investigated by

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Fröhlich, Kaplansky [109], and Cordes [39] (a deﬁnitive exposition in [162, pp. 95– 99]). Kaplansky introduced the concept of a radical of any ﬁeld F . This is the set of all a ∈ F˙ for which the quaternion4 algebra (a, b/F ) splits for all b ∈ F˙ . Equivalently, the radical is the intersection of the value sets of all one-fold Pﬁster forms over F . Kaplansky studied quadratic form theory over ﬁelds F with Q = 2, and for formally real ﬁelds, these are exactly the ﬁelds with the index of the radical in F˙ equal to two. The higher radicals were introduced in [241], and Kaplansky’s results were generalized to the context of n-fold Pﬁster forms in [225]. Cordes [40] proved that, if Q(F ) = 4, the ETC holds for F , and Kula [136] proved that, if Q(F ) = 8, the ETC holds for F . Another important piece of evidence for the ETC comes from the work of Jacob and Ware [105]. They proved that, for a ﬁeld F with ﬁnite square class group, if W(F ) is a ﬁnite direct√product of Witt rings, then the Witt ring W(K) of any quadratic extension K = F( a) is a ﬁnite direct product of abstract Witt rings describable in terms of factors of W(F ). In particular, when √ the Witt ring W(F ) is of elementary type, then for any a ∈ F˙ the Witt ring W(F( a)) is of elementary type. Kula [137] studied the number of Witt rings for ﬁelds with a given number of square classes and its asymptotics. Scharlau [206] deﬁned the generating function for a ﬁeld F with ﬁnite group of square classes as the power series 1 + i>0 ci xi , where ci is the number of equivalence classes of n-dimensional quadratic forms over F . If the Witt ring W(F ) is of elementary type, he proved that the series is a rational function with the only pole at x = 1. Kula [135] proved that this is so for an arbitrary ﬁeld F with ﬁnite square class group. Kula, Marshall, and Sładek [138] investigated the structure of direct limits of torsion-free abstract Witt rings. Various ring properties of Witt rings of ﬁelds and abstract Witt rings have been studied, such as units [154,164], level and sums of squares [153,155], automorphisms [146], annihilating polynomials [97, 151, 152], primary ideals, ideal class groups and Picard groups [78–80], and projectivity of W(K) over W(F ) for a ﬁeld extension K/F [81]. The best source for abstract Witt rings is Marshall’s book [162]. For a comprehensive survey of work related to elementary type conjecture, citing 117 papers, see Marshall [165]. For an approach to an axiomatization of quadratic form theory in terms of the isometry relation on binary forms, see Dickmann and Miraglia [59, 60]. 10.2. Reduced theory of quadratic forms For a formally real ﬁeld F , there exists a complete theory of quadratic forms relative to a ﬁxed preordering of the ﬁeld. Here, a preordering in F is a proper subset T of F closed with respect to addition and multiplication and satisfying F˙ 2 ⊆ T . Any 4 Actually, Kaplansky avoids quaternion algebras and uses Hilbert symbols instead. As to the term radical of ﬁeld, Kaplansky says “Since any conventional concept of radical is ridiculous for a ﬁeld, I trust there will be no confusion.”

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preordering T in F is contained in at least one ordering P of F (i.e., a preordering P satisfying additionally P ∪ −P = F ). The set of all orderings of F extending a given preordering one shows T is denoted as X/T . Using the Kuratowski–Zorn Lemma, that T = P, where as P ranges over X/T . In case of T = F˙ 2 , this recaptures the Artin–Schreier characterization of totally positive elements in an ordered ﬁeld. A T -form is just a ﬁnite sequence f = (a1 , . . . , an ) of elements in F˙ and at each ordering P ∈ X/T , we have the signature sgnP f . Two forms f and g are said to be T -isometric if they have the same dimension n and have the same signatures with respect to any ordering P extending T . With these deﬁnitions, one can now follow the classical quadratic form theory pattern over ﬁelds and deﬁne the counterparts of all the classical concepts as WT (F ). In the special 2 including the Witt ring, denoted case, when T = F˙ , it may be shown that WT (F ) ∼ = W(F )/Wt (F ), and this ring is said to be the reduced Witt ring of the ﬁeld F . The set X = X/ F˙ 2 of all orderings of the ﬁeld F can be viewed as a topological space with the induced Zariski topology from the prime spectrum of the Witt ring W(F ). Alternatively, the Harrison sets H(a) = {P ∈ X : a ∈ P} ,

a ∈ F˙

form a subbasis. X is a Boolean space and for each preordering T , the set X/T is closed, hence also a Boolean space. We now consider the ring C(X/T, Z) of continuous functions from the space X/T to Z (with discrete topology) and the map cT : WT (F ) → C(X/T, Z),

cT (f ) = fˆ ,

where fˆ : X/T → Z, fˆ(P) = sgnP f . One checks that cT is an injective ring homomorphism and it is of great interest to describe precisely the image of cT . The functions in the image are said to be represented over T and the representation problem asks for a description of represented functions. A crucial role in solving the representation problem is played by a special type of preorderings called fans. A preordering T is a fan when, for any subset S of F , if −1 ∈ / S and S˙ is a subgroup of F˙ , then S is a preordering in F . The ultimate result reads as follows. Theorem 10.2 (Representation Theorem, Becker and Bröcker [19]). Let T0 be a preordering in F . Then a function f ∈ C(X/T0 ), Z) is represented over T0 iff for every fan T ⊇ T0 of ﬁnite index in F ,

f(P) ≡ 0

(mod |X/T |).

P∈X/T

A comprehensive survey of the reduced theory, the relations to valuation theory, and detailed proofs are provided in Lam’s notes [142]. There are 164 references there, but Lam says the list is incomplete and one should consult the bibliography in his survey on ordered ﬁelds [141] where there are 223 references.

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11. Historical Since the time of Fermat, eminent mathematicians were always interested in phenomena connected with quadratic forms. However, quadratic forms were considered usually over the integers or over the ﬁelds R and C. The equivalence problem over integers is extraordinarily difﬁcult and over R and C it is trivial. Hence, ﬁnding criteria for equivalence of quadratic forms was not the central problem. This changed at the end of 19th century when Minkowski [178] solved the equivalence problem for quadratic forms over the rationals, and Hilbert included in his list of 23 problems the following eleventh problem.5 Our present knowledge of the theory of quadratic number ﬁelds puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefﬁcients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefﬁcients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefﬁcients. Thus, Hilbert asks for solution of the representation problem over algebraic number ﬁelds and over rings of algebraic integers. As we know, in the case of ﬁelds, a solution of the representation problem implies a solution of the equivalence problem; hence, Hilbert’s question addresses the more general problem. Hilbert’s high opinion on the present knowledge is certainly, at least in part, based on the results of his 1899 paper [91]. He presented there a complete theory of the Hilbert symbol in relative quadratic extensions of nonreal algebraic number ﬁelds with odd class number. For such a ﬁeld K, he proved ([91, Section 44, Satz 66]) that the equation ax2 + by2 = 1 has a solution in K if and only if for every prime ideal p of K we have (a, b/p) = 1. This condition is equivalent to the solvability of the congruence ax2 + by2 ≡ 1 (mod pn ) in integers x, y in K for all exponents n. Hilbert does not say so, but the latter is equivalent to the solvability of the equation ax2 + by2 = 1 in the p-adic completions of the ﬁeld K. Thus, Hilbert’s result is a prototype of a local–global principle: the equation ax2 + by2 = 1 is solvable in the ﬁeld K if and only if it is solvable in all completions of K. Five papers of Hasse [86–90] give a complete solution to the algebraic part of Hilbert’s problem XI. Hasse does not mention the Hilbert problem at all, but there is no doubt that his results form a theory of quadratic forms over number ﬁelds, as demanded by Hilbert. Hasse shows that two quadratic forms are equivalent over a number ﬁeld F if and only if they are equivalent over all completions Fp of F (where p runs through all primes of F , including inﬁnite primes). A similar principle holds for the representation problem: a quadratic form over number ﬁeld F represents an element of the ﬁeld F if and only if it represents that element over all completions Fp of F . Thus, the solutions of the equivalence and representation problems over the number ﬁeld F depend exclusively on solutions of these problems in the local ﬁelds, Fp . For local ﬁelds, Hasse found a complete solution of both problems in 5 Translation taken from BAMS 8 (1902), 437–479.

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principle equivalent to what follows from Theorem 6.1 before. In this way, the Hasse local–global principle becomes an effective tool for solving the equivalence and representation problems for quadratic forms over algebraic number ﬁelds. Another important event in the 1920s was Artin’s solution of Hilbert’s 17th problem titled by Hilbert “Expression of deﬁnite forms by squares.” For a ﬁeld F , let S(F ) denote the set of all sums of squares of elements of F , and let T(F ) be the set of all elements of F , that is, nonnegative at every ordering of F . Further, for the rational function ﬁeld K = F(X1 , . . . , Xn ) over an ordered ﬁeld F , let D(K) be the set of positive semideﬁnite elements of K. Thus, D(K) consists of rational functions f(X1 , . . . , Xn ) ∈ K with f(a1 , . . . , an ) non-negative in F for all n-tuples a1 , . . . , an ∈ F in the domain of f . Clearly, S(K) ⊆ T(K)

and

S(K) ⊆ D(K).

Hilbert’s 17th problem asked whether S(K) = D(K) when F = R. Thus, the question was whether every positive semideﬁnite rational function over R can be written as the sum of squares of rational functions over R. Obviously, this question reduces to representing positive semideﬁnite polynomials over R as sums of squares of rational functions over R. Already Hilbert knew that there are semideﬁnite polynomials, which cannot be written as sums of squares of polynomials, but the ﬁrst explicit example of such a polynomial was found only in 1967 by T. Motzkin, f = 1 + X2 Y 4 + X4 Y 2 − 3X2 Y 2 . Hilbert also knew that the answer to his question is in the positive when n = 2. Hilbert’s problem was solved by E. Artin in 1927. Two main results in his paper [8] assert that, ﬁrst, S(K) = T(K) for all ﬁelds K [with the convention that T(K) = K for a nonreal ﬁeld K], and second, T(K) = D(K) for any rational function ﬁeld K = F(X1 , . . . , Xn ), where F is a number ﬁeld with exactly one ordering, for example, F = R. Thus, S(K) = T(K) = D(K) for any K = F(X1 , . . . , Xn ), where F is a subﬁeld of R with a unique ordering. Artin’s solution was based on the theory of formally real ﬁelds founded by Artin and Schreier in [9]. Brauer [22] comments on these results as follows: “His method was as remarkable as the result. It was perhaps the ﬁrst triumph of what is sometimes called abstract algebra.” After Hasse and Artin, the third important paper of that time, and perhaps the most inﬂuential paper on quadratic forms, is the 1937 Ernst Witt paper [240]. He introduced the geometric language of “metric spaces,” that is, vector spaces with quadratic (and bilinear) forms, and proved a series of theorems which became a standard foundation of quadratic form theory. Thus, he proves the orthogonal complement theorem, the existence of orthogonal bases, the cancellation theorem (Satz 4), the theorem on decomposition of isotropic forms into a direct sum of hyperbolic and anisotropic forms. Then he proves that classes of similar forms form a commutative ring (Satz 6) and proves the chain equivalence theorem (Satz 7). In the second part of the paper, he introduces the Clifford algebra of an arbitrary quadratic form (not just sum of squares) and uses it to prove that the Hasse algebra of a quadratic form is a welldeﬁned equivalence invariant. It is amusing that Witt used for this Clifford algebras

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and did not notice that his chain equivalence theorem (Satz 7) does the job effortlessly. Then he proceeds to a series of classiﬁcation results of which the ﬁrst (Satz 11) is Theorem 4.1 above. In the third part of the paper Witt gives new, concise and lucid proof of the Hasse principle. In particular, he deduces the local–global principle for equivalence from the principle for the representation problem using his cancellation theorem. The ﬁnal section of the paper solves the classiﬁcation problem over real algebraic function ﬁelds in one variable. While Witt’s paper is of utmost importance, it is interesting and surprising that at some points it was anticipated by a 1907 Dickson paper [61]. As reported by Scharlau [207], Dickson not only found the diagonalizability of quadratic forms over ﬁelds of characteristic = 2 but also proved the cancellation theorem! It is a melancholy thought to realize that it took almost 100 years to recognize Dickson’s contribution. After Witt’s paper, there has been very little activity in quadratic form theory with sporadic, albeit important, publications such as Kaplansky’s paper [108] mentioned in Section 7.2 and Springer [219, 220]. Then in 1965, the ﬁrst paper of Pﬁster [190] appeared with the solution of the level problem for ﬁelds. In his next papers [191, 192], the local–global principle (Theorem 3.3) for torsion forms was proved and the quantitative solution of Hilbert’s 17th problem was obtained (Theorem 8.5). These results attracted the attention of many researchers and were the starting point of continual activity in quadratic form theory over ﬁelds. The progress was surveyed by Pﬁster himself in at least ﬁve surveys [193–196, 199]. Also Scharlau [207] gives an overview, which concentrates on “some aspects neglected in the existing literature.”

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[222] M. Szyjewski, The ﬁfth invariant of quadratic forms. (Russian), Dokl. Akad. Nauk SSSR 308 (1989), 542–545. Translation in: Soviet Math. Doklady 40 (1990), 355–358. [223] M. Szyjewski, On the Witt ring of a relative projective line, Coll. Math. 75 (1998), 53–78. [224] K. Szymiczek, Quadratic forms over ﬁelds with ﬁnite square class number, Acta Arith. 28 (1975), 195–221. [225] K. Szymiczek, Generalized Hilbert ﬁelds, J. Reine Angew. Math. 329 (1981), 58–65. [226] K. Szymiczek, Matching Witts locally and globally, Math. Slovaca 41 (1991), 315–330. [227] K. Szymiczek, Witt equivalence of global ﬁelds, Commun. Algebra 19 (1991), 1125–1149. [228] K. Szymiczek, Algebra, Logic and Applications, vol. 7: Bilinear Algebra. An introduction to the algebraic theory of quadratic forms, Gordon and Breach Science Publishers, Amsterdam, 1997. [229] K. Szymiczek, Hilbert-symbol equivalence of number ﬁelds, Tatra Mount. Math. Publ. 11 (1997), 7–16. [230] K. Szymiczek,Acharacterization of tame Hilbert-symbol equivalence,Acta Math. Inf. Univ. Ostrav. 6 (1998), 191–201. [231] K. Szymiczek, Tame equivalence and wild sets, Semigroup Forum 60 (2000), 260–270. [232] K. Szymiczek, Ten problems on quadratic forms, Acta Math. Inf. Univ. Ostrav. 10 (2002), 133–143. [233] C.C. Tsen, Zur Stufentheorie der quasialgebraisch-Abgeschlossenheit kommutativer Körper, J. Chin. Math. Soc. 1 (1936), 81–92. [234] A. Vishik, On dimensions of quadratic forms, Dokl. Akad. Nauk 373 (2000), no. 4, 445–447. [235] A. Vishik, Fields of u-invariant 2r + 1. Preprint No. 229, October 2006. Linear Algebraic Groups and Related Structures preprint server (http://www.math.uni-bielefeld.de/LAG/). [236] V. Voevodsky, On 2−torsion in motovic cohomology. Preprint 2001. Available at K-theory Preprint Archives http://www.math.uiuc.edu/K-theory/0502/ [237] A.R. Wadsworth, Similarity of quadratic forms and isomorphism of their function ﬁelds, Trans. Am. Math. Soc. 208 (1975), 352–358. [238] A.R. Wadsworth, Merkurjev’s elementary proof of Merkurjev’s theorem, Contemp. Math. 55 (1986), 741–776. [239] W.C. Waterhouse, Scaled trace forms over number ﬁelds, Arch. Math. 47 (1986), 229–231. [240] E. Witt, Theorie der quadratischen Formen in beliebigen Körpern, J. Reine Angew. Math. 176 (1937), 31–44. [241] J.L. Yucas, Quadratic forms and radicals of ﬁelds, Acta Arith. 39 (1981), 313–322.

Uncited References [1] T.C. Craven, Orderings, valuations, and hermitian forms over *-ﬁelds, Proc. Symp. Pure Math. 58, Part II (1995), 149–160. [2] R. Elman, A. Prestel, Reduced stability of the Witt ring of a ﬁeld and its Pythagorean closure, Am. J. Math. 106 (1984), 1237–1260. [3] R.W. Fitzgerald, J. Yucas, Combinatorial techniques and abstract Witt rings, I. J. Algebra 114 (1988), 40–52. [4] R.W. Fitzgerald, J. Yucas, Combinatorial techniques and abstract Witt rings, II. Rocky Mountain J. Math. 19 (1989), 687–708. [5] U. Jannsen, R. Sujatha, Levels of function ﬁelds of surfaces over number ﬁelds, J. Algebra 251 (2002), no. 1, 350–357. [6] I. Kaplansky, J. Shaker, Abstract quadratic forms, Can. J. Math. 21 (1969), 1218–1233. [7] M.-A. Knus, T.Y. Lam, D.B. Shapiro, J.-P. Tignol, Discriminants of involutions on biquaternion algebras, Proc. Symp. Pure Math. 58, Part II (1995), 279–303. [8] T.Y. Lam, The Algebraic Theory of Quadratic Forms, W.A. Benjamin, Addison-Wesley, Reading, Mass., 1973. Second printing with revisions, 1980.

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[9] T.Y. Lam, D.B. Leep, J.-P. Tignol, Biquaternion algebras and quartic extensions, Math. Ann. 62 (1991), 272–285. [10] F. Lorenz, Lecture Notes in Mathematics, vol. 130: Quadratische Formen über Körpern, SpringerVerlag, Berlin-Heidelberg-New York, 1970, 2007. [11] P. Mammone, D.B. Shapiro, The Albert quadratic form for an algebra of degree four, Proc. Am. Math. Soc. 105 (1989), 525–530. [12] A. Prestel, Lecture Notes in Mathematics, vol. 1093: Lectures on Formally Real Fields, SpringerVerlag, Berlin Heidelberg New York Tokyo, 1984. [13] D.B. Shapiro, Products of sums of squares, Expos. Math. 2 (1984), 235–261.

Crossed Complexes and Higher Homotopy Groupoids as Noncommutative Tools for Higher Dimensional Local-to-Global Problems∗ Ronald Brown School of Computer Science, Bangor University, Bangor, Gwynedd LL57 1UT, U.K. E-mail: [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Crossed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The fundamental groupoid on a set of base points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The search for higher homotopy groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Why crossed complexes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Why cubical ω-groupoids with connections? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. First main aim of the work: higher Homotopy van Kampen theorems . . . . . . . . . . . . . . . . . . . . 9. The fundamental cubical ω-groupoid ρX∗ of a ﬁltered space X∗ . . . . . . . . . . . . . . . . . . . . . . . 10. Collapsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Partial boxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Thin elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Sketch proof of the HHvKT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Tensor products and homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Free crossed complexes and free crossed resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16. Classifying spaces and the homotopy classiﬁcation of maps . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 87 89 92 98 100 101 102 103 104 106 107 108 108 110 112 113

∗ This is a revised version (2008) of a paper published in Fields Institute Communications 43 (2004) 101–130, which was an extended account of a lecture given at the meeting on “Categorical Structures for Descent, Galois Theory, Hopf algebras and semiabelian categories,” Fields Institute, September 23–28, 2002. The author is grateful for support from the Fields Institute and a Leverhulme Emeritus Research Fellowship, 2002–2004 and to M. Hazewinkel for helpful comments on a draft.

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00203-9 83

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17. Relation with chain complexes with a groupoid of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Crossed complexes and simplicial groups and groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. Other homotopy multiple groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Conclusion and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114 116 117 118 119

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Introduction An aim is to give a survey and explain the origins of results obtained by R. Brown and P.J. Higgins and others over the years 1974–2008, and to point to applications and related areas. These results yield an account of some basic algebraic topology on the border between homology and homotopy; it differs from the standard account through the use of crossed complexes, rather than chain complexes, as a fundamental notion. In this way, one obtains comparatively quickly1 not only classical results such as the Brouwer degree and the relative Hurewicz theorem but also noncommutative results on second relative homotopy groups, as well as higher dimensional results involving the fundamental group through its actions and presentations. A basic tool is the fundamental crossed complex X∗ of the ﬁltered space X∗ , which in the case X0 is a singleton is fairly classical; when applied to the skeletal ﬁltration of a CW -complex X, gives a more powerful version of the usual cellular chains of the universal cover of X because it contains non-Abelian information in dimensions 1 and 2 and has good realization properties. It also gives a replacement for singular chains by taking X to be the geometric realization of a singular complex of a space. One of the major results is a homotopy classiﬁcation theorem (4.1.9), which generalizes a classical theorem of Eilenberg–Mac Lane, though this does require results on geometric realizations of cubical sets. A replacement for the excision theorem in homology is obtained by using cubical methods to prove a Higher Homotopy van Kampen Theorem (HHvKT)2 for the fundamental crossed complex functor on ﬁltered spaces. This theorem is a higher dimensional version of the van Kampen Theorem (vKT) on the fundamental group of a space with base point, [1]3 , which is a classical example of a noncommutative local-to-global theorem, and is the initial motivation for the work described here. The vKT determines completely the fundamental group π1 (X, x) of a space X with base point, which is the union of open sets U, V whose intersection is path-connected and contains the base point x; the “local information” is on the morphisms of fundamental groups induced by the inclusions U ∩ V → U, U ∩ V → V . The importance of this result reﬂects the importance of the fundamental group in algebraic topology, algebraic geometry, complex analysis, and many other subjects. Indeed, the origin of the fundamental group was in Poincaré’s work on monodromy for complex variable theory. Essential to this use of crossed complexes, particularly for conjecturing and proving local-to-global theorems, is a construction of a cubical higher homotopy groupoid, with properties described by an algebra of cubes. There are applications to localto-global problems in homotopy theory, which are more powerful than available by 1 This comparison is based on the fact that the methods do not require singular homology or simplicial

approximation. 2 We originally called this a generalized van Kampen Theorem, but this new term was suggested in 2007 by Jim Stasheff. 3 An earlier version for simplicial complexes is due to Seifert.

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purely classical tools while shedding light on those tools. It is hoped that this account will increase the interest in the possibility of wider applications of these methods and results since homotopical methods play a key role in many areas. Background in higher homotopy groups Topologists in the early part of the 20th century were well aware that • the noncommutativity of the fundamental group was useful in geometric applications; • for path-connected X, there was an isomorphism H1 (X) ∼ = π1 (X, x)ab ; • the Abelian homology groups Hn (X) existed for all n 0. Consequently, there was a desire to generalize the noncommutative fundamental group to all dimensions. ˇ In 1932, Cech submitted a paper on higher homotopy groups πn (X, x) to the ICM at Zurich, but it was quickly proved that these groups were Abelian for n 2, and ˇ on these grounds, Cech was persuaded to withdraw his paper so that only a small paragraph appeared in the Proceedings [2]. We now see the reason for this commutativity as the result (Eckmann–Hilton) that a group internal to the category of groups is just an Abelian group. Thus, since 1932, the vision of a noncommutative higher dimensional version of the fundamental group has been generally considered to be a mirage. Before we go back to the vKT, we explain in the next section how nevertheless work on crossed modules did introduce noncommutative structures relevant to topology in dimension 2. Work of Hurewicz, [3], led to a strong development of higher homotopy groups. The fundamental group still came into the picture with its action on the higher homotopy groups, which I once heard J.H.C. Whitehead remark (1957) was especially fascinating for the early workers in homotopy theory. Much of Whitehead’s work was intended to extend to higher dimensions the methods of combinatorial group theory of the 1930s, hence the title of his papers were as follows: “Combinatorial homotopy, I, II” [4, 5]. The ﬁrst of these two papers has been very inﬂuential and is part of the basic structure of algebraic topology. It is the development of work of the second paper, which we explain here. The paper by Whitehead on “Simple homotopy types” [6], which deals with higher dimensional analogs of Tietze transformations, has a ﬁnal section using crossed complexes. We refer to this again later in Section 15. It is hoped also that this survey will be useful background to work on the van Kampen Theorem for diagrams of spaces in [7], which uses a form of higher homotopy groupoid that is in an important sense much more powerful than that given here since it encompasses n-adic information; however, current expositions are still restricted to the reduced (one base point) case, the proof uses advanced tools of algebraic topology, and the result was suggested by the work exposed here.

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1. Crossed modules In the years 1941–1950, Whitehead developed work on crossed modules to represent the structure of the boundary map of the relative homotopy group π2 (X, A, x) → π1 (A, x),

(1)

in which both groups can be noncommutative. Here is the deﬁnition he found. A crossed module is a morphism of groups μ : M → P together with an action (m, p) → mp of the group P on the group M satisfying the following two axioms: CM1) μ(mp ) = p−1 (μm)p CM2) n−1 mn = mμn for all m, n ∈ M, p ∈ P. Standard algebraic examples of crossed modules are as follows: (i) an inclusion of a normal subgroup, with action given by conjugation; (ii) the inner automorphism map χ : M → Aut M, in which χm is the automorphism n → m−1 nm; (iii) the zero map M → P where M is a P-module; (iv) an epimorphism M → P with kernel contained in the center of M. Simple consequences of the axioms for a crossed module μ : M → P are as follows: 1.1. Im μ is normal in P. 1.2. Ker μ is central in M and is acted on trivially by Im μ so that Ker μ inherits an action of M/ Im μ. Another important algebraic construction is the free crossed P-module ∂ : C(ω) → P determined by a function ω : R → P, where P is a group and R is a set. The group C(ω) is generated by elements (r, p) ∈ R × P with the relations (r, p)−1 (s, q)−1 (r, p)(s, qp−1 (ωr)p); the action is given by (r, p)q = (r, pq); and the boundary morphism is given by ∂(r, p) = p−1 (ωr)p, for all (r, p), (s, q) ∈ R × P. A major result of Whitehead was Theorem W. [5]. If the space X = A ∪ {e2r }r∈R is obtained from A by attaching 2-cells by maps fr : (S 1 , 1) → (A, x), then the crossed module of (1) is isomorphic to the free crossed π1 (A, x)-module on the classes of the attaching maps of the 2-cells. Whitehead’s proof, which stretched over three papers, 1941–1949, used transversality and knot theory—an exposition is given in [8]. Mac Lane and Whitehead [9]

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used this result as part of their proof that crossed modules capture all homotopy 2-types (they used the term “3-types”). The title of the paper in which the ﬁrst intimation of Theorem W appeared was “On adding relations to homotopy groups” [10]. This indicates a search for higher dimensional vKTs. The concept of free crossed module gives a noncommutative context for chains of syzygies. The latter idea, in the case of modules over polynomial rings, is one of the origins of homological algebra through the notion of free resolution. Here is how similar ideas can be applied to groups. Pioneering work here, independent of Whitehead, was by Peiffer [11] and Reidemeister [12]. See [13] for an exposition of these ideas. Suppose P = X | ω is a presentation of a group G so that X is a set of generators of G and ω : R → F(X) is a function, whose image is called the set of relators of the presentation. Then we have an exact sequence i

φ

1 −→ N(ωR) −→ F(X) −→ G −→ 1, where N(ωR) is the normal closure in F(X) of the set ωR of relators. The above work of Reidemeister, Peiffer, and Whitehead showed that to obtain the next level of syzygies, one should consider the free crossed F(X)-module ∂ : C(ω) → F(X) since this takes into account the operations of F(X) on its normal subgroup N(ωR). Elements of C(ω) are a kind of “formal consequences of the relators” so that the relation between the elements of C(ω) and those of N(ωR) is analogous to the relation between the elements of F(X) and those of G. It follows from the rules for a crossed module that the kernel of ∂ is a G-module, called the module of identities among relations, and sometimes written as π(P); there is considerable work on computing it [13–17, 126]. By splicing to ∂ a free G-module resolution of π(P), one obtains what is called a free crossed resolution of the group G. We explain later (Proposition 15.3) why these resolutions have better realization properties than the usual resolutions by chain complexes of G-modules. They are relevant to the Schreier extension theory [18]. This notion of using crossed modules as the ﬁrst stage of syzygies in fact represents a wider tradition in homological algebra, in the work of Frölich and Lue [19, 20]. Crossed modules also occurred in other contexts, notably in representing elements of the cohomology group H 3 (G, M) of a group G with coefﬁcients in a G-module M [21] and as coefﬁcients in Dedecker’s theory of non-Abelian cohomology [22]. The notion of free crossed resolution has been exploited by Huebschmann [23–25] to represent cohomology classes in H n (G, M) of a group G with coefﬁcients in a G-module M and also to calculate with these. The HHvKT can make it easier to compute a crossed module arising from some topological situation, such as an induced crossed module [26, 27], or a coproduct crossed module [28], than the cohomology class in H 3 (G, M) the crossed module represents. To obtain information on such a cohomology element, it is useful to work with a small free crossed resolution of G, and this is one motivation for developing methods for calculating such resolutions. However, it is not so clear what a calculation of such a cohomology element would amount to although it is interesting to know

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whether the element is nonzero or what its order is. Thus, the use of algebraic models of cohomology classes may yield easier computations than the use of cocycles, and this somewhat inverts traditional approaches. Since crossed modules are algebraic objects generalizing groups, it is natural to consider the problem of explicit calculations by extending techniques of computational group theory. Substantial work on this has been done by C.D. Wensley using the program GAP [29, 30].

2. The fundamental groupoid on a set of base points A change in prospects for higher order noncommutative invariants was suggested by Higgins’ paper [31] and leading to work of the writer published in 1967 [32]. This showed that the vKT could be formulated for the fundamental groupoid π1 (X, X0 ) on a set X0 of base points, thus enabling computations in the nonconnected case, including those in Van Kampen’s original paper [1]. This successful use of groupoids in dimension 1 suggested the question of the use of groupoids in higher homotopy theory and, in particular, the question of the existence of higher homotopy groupoids. To see how this research programme could progress, it is useful to consider the statement and special features of this generalized vKT for the fundamental groupoid. If X0 is a set, and X is a space, then π1 (X, X0 ) denotes the fundamental groupoid on the set X ∩ X0 of base points. This allows the set X0 to be chosen in a way appropriate to the geometry. For example, if the circle S 1 is written as the union of two semicircles E+ ∪ E− , then the intersection {−1, 1} of the semicircles is not connected, so it is not clear where to take the base point. Instead, one takes X0 = {−1, 1}, and so has two base points. This ﬂexibility is very important in computations, and this example of S 1 was a motivating example for this development. As another example, you might like to consider the difference between the quotients of the actions of Z2 on the group π1 (S 1 , 1) and on the groupoid π1 (S 1 , {−1, 1}), where the action is induced by complex conjugation on S 1 . Relevant work on orbit groupoids has been developed by Higgins and Taylor [33, 34], (under useful conditions, the fundamental groupoid of the orbit space is the orbit groupoid of the fundamental groupoid [35, 11.2.3]). Consideration of a set of base points leads to the following theorem (Theorem 2.1): Theorem 2.1. [32] Let the space X be the union of open sets U, V with intersection W , and let X0 be a subset of X meeting each path component of U, V, W. Then, (C) (connectivity) X0 meets each path component of X and (I) (isomorphism) the diagram of groupoid morphisms induced by inclusions π1 (W, X0 )

i

j

j

π1 (V, X0 )

/ π1 (U, X0 )

i

is a pushout of groupoids.

/ π1 (X, X0 )

(2)

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From this theorem, one can compute a particular fundamental group π1 (X, x0 ) using combinatorial information on the graph of intersections of path components of U, V, W, but for this, it is useful to develop the algebra of groupoids. Notice two special features of this result, which are as follows. (i) The computation of the invariant you may want, a fundamental group, is obtained from the computation of a larger structure, and so part of the work is to give methods for computing the smaller structure from the larger one. This usually involves noncanonical choices, for example, that of a maximal tree in a connected graph. The work on applying groupoids to groups gives many examples of this [31, 32, 36, 37]. (ii) The fact that the computation can be done is surprising in two ways: (a) the fundamental group is computed precisely, even though the information for it uses input in two dimensions, namely 0 and 1. This is contrary to the experience in homological algebra and algebraic topology, where the interaction of several dimensions involves exact sequences or spectral sequences, which give information only up to extension, and (b) the result is a noncommutative invariant, which is usually even more difﬁcult to compute precisely. The reason for the success seems to be that the fundamental groupoid π1 (X, X0 ) contains information in dimensions 0 and 1, and so can adequately reﬂect the geometry of the intersections of the path components of U, V, W and of the morphisms induced by the inclusions of W in U and V . This suggested the question of whether these methods could be extended successfully to higher dimensions. Part of the initial evidence for this quest was the intuitions in the proof of this groupoid vKT, which seemed to use three main ideas to verify the universal property of a pushout for diagram (2). So, suppose given morphisms of groupoids fU , fV from π1 (U, X0 ), π1 (V, X0 ) to a groupoid G, satisfying fU i = fV j. We have to construct a morphism f : π1 (X, X0 ) → G such that fi = fU , fj = fV and prove f is unique. We concentrate on the construction. • One needs a “deformation,” or “ﬁlling,” argument: given a path a : (I, I˙ ) → (X, X0 ), one can write a = a1 + · · · + an where each ai maps into U or V , but ai will not necessarily have end points in X0 . So one has to deform each ai to ai in U, V , or W , using the connectivity condition so that each ai has end points in X0 , and a = a1 + · · · + an is well deﬁned. Then, one can construct using fU or fV an image of each ai in G and hence of the composite, called F(a) ∈ G, of these images. Note that we subdivide in X and then put together again in G (this uses the condition fU i = fV j to prove that the elements of G are composable), and this part can be summarized as follows: • Groupoids provided a convenient algebraic inverse to subdivision. Note that the usual exposition in terms only of the fundamental group uses loops, that is, paths which start and ﬁnish at the same point. An appropriate analogy is that if one goes on a train journey from Bangor and back to Bangor, one usually wants to

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stop off at intermediate stations; this breaking and continuing a journey is better described in terms of groupoids rather than groups. Next, one has to prove that F(a) depends only on the class of a in the fundamental groupoid. This involves a homotopy relative end points h : a b, considered as a map I 2 → X; subdivide h as h = [hij ] so that each hij maps into U, V , or W ; deform h to h = [hij ] (keeping in U, V, W) so that each hij maps the vertices to X0 and so determines a commutative square4 in one of π1 (Q, X0 ) for Q = U, V, W. Move these commutative squares over to G using fU , fV and recompose them (this is possible again because of the condition fU i = fV j), noting that • in a groupoid, any composition of commutative squares is commutative. Here, a “big” composition of commutative squares is represented by a diagram such as •

/•

/•

/•

/•

/•

/•

•

/ •

/ •

/ •

/ •

/ •

/ •

•O

/ •O

/ •

/ •

/ •

/ •

/ •

•

/ •

/ •

/ •

/ •

/ •

/ •

(3)

and one checks that if each individual square is commutative, so also is the boundary square (later called a 2-shell) of the compositions of the boundary edges. ⎛ 4 We need the notion of commutative square in a category C. This is a quadruple ⎝a

c

⎞

d ⎠ of arrows b in C, called “edges” of the square such that ab = cd, that is, these compositions are deﬁned and agreed. The commutative squares in C form a double category C in that they compose “vertically” ⎛ ⎝a

and “horizontally”

⎛ ⎝a

c b

c b

⎞

⎛

d ⎠ ◦1 ⎝a

⎞

⎛

d ⎠ ◦2 ⎝d

b e

c b

⎞

⎛

d ⎠ = ⎝aa

⎞

⎛

f ⎠ = ⎝a

c e

cc bb

⎞ dd ⎠

⎞ f⎠

This notion of C was deﬁned by C. Ehresmann in papers and in [38]. Note the obvious geometric conditions for these compositions to be deﬁned. Similarly, one has geometric conditions for a rectangular array (cij ), 1 i m, 1 j n, of commutative squares to have a well-deﬁned composition, and then their “multiple composition,” written as [cij ], is also a commutative square, whose edges are compositions of the “edges” along the outside boundary of the array. It is easy to give formal deﬁnitions of all this.

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Two opposite sides of the composite commutative square in G so obtained are identities because h was a homotopy relative to end points, and the other two sides are F(a), F(b). This proves that F(a) = F(b) in G. Thus, the argument can be summarized as follows: a path or homotopy is divided into small pieces, then deformed so that these pieces can be packaged and moved over to G, where they are reassembled. There seems to be an analogy with the processing of an e-mail. Notable applications of the groupoid theorem were: (i) to give a proof of a formula in van Kampen’s paper of the fundamental group of a space, which is the union of two connected spaces with non connected intersection, see [35, 8.4.9]; and (ii) to show the topological utility of the construction by Higgins [36] of the groupoid f∗ (G) over Y0 induced from a groupoid G over X0 by a function f : X0 → Y0 . (Accounts of these with the notation Uf (G) rather than f∗ (G) are given in [35, 36].) This latter construction is regarded as a “change of base,” and analogs in higher dimensions yielded generalizations of the relative Hurewicz theorem and of the Theorem W, using induced modules and crossed modules. There is another approach to the vKT, which goes via the theory of covering spaces and the equivalence between covering spaces of a reasonable space X and functors π1 (X) → Set [35]. See for example [39] for an exposition of the relation with traditional Galois theory and [40] for a modern account in which Galois groupoids make an essential appearance. The paper [41] gives a general formulation of conditions for the theorem to hold in the case X0 = X in terms of the map U V → X being an “effective global descent morphism” (the theorem is given in the generality of lextensive categories). This work has been developed for toposes [42]. Analogous interpretations for higher dimensional vKTs are not known. The justiﬁcation of the breaking of a paradigm in changing from groups to groupoids is several fold: the elegance and power of the results; the increased linking with other uses of groupoids [43]; and the opening out of new possibilities in higher dimensions, which allowed for new results and calculations in homotopy theory and suggested new algebraic constructions. The important and extensive work of Charles Ehresmann in using groupoids in geometric situations (bundles, foliations, germes, . . .) should also be stated (see his collected works of which [44] is volume 1 and a survey [45]).

3. The search for higher homotopy groupoids Contemplation of the proof of the groupoid vKT in the last section suggested that a higher dimensional version should exist, though this version amounted to an idea of a proof in search of a theorem. Further evidence was the proof by J.F. Adams of the cellular approximation theorem given in [35]. This type of subdivision argument failed to give algebraic information apparently because of a lack of an appropriate higher homotopy groupoid, that is a gadget to capture what might be the underlying “algebra of cubes.” In the end, the results exactly encapsulated this intuition.

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One intuition was that in groupoids, we are dealing with a partial algebraic structure,5 in which composition is deﬁned for two arrows if and only if the source of one arrow is the target of the other. This seems to generalize easily to directed squares, in which two such are composable horizontally if and only if the left-hand side of one is the right-hand side of the other (and similarly vertically). However, the formulation of a theorem in higher dimensions required speciﬁcation of the topological data, the algebraic data, and of a functor : (topological data) → (algebraic data), which would allow the expression of these ideas for the proof. Experiments were made in the years 1967–1973 to deﬁne some functor from spaces to some kind of double groupoid, using compositions of squares in two directions, but these proved abortive. However, considerable progress was made in work with Chris Spencer in 1971–1973 on investigating the algebra of double groupoids [47] and on showing a relation to crossed modules. Further evidence was provided when it was found, [48], that group objects in the category of groupoids (or groupoid objects in the category of groups, either of which are often now called “2-groups”) are equivalent to crossed modules and, in particular, are not necessarily commutative objects. It turned out that this result was known to the Grothendieck school in the 1960s, but not published. We review next a notion of double category, which is not the most general but is appropriate in many cases. It was called an edge symmetric double category in [49]. In the ﬁrst place, a double category, K, consists of a triple of category structures (K2 , K1 , ∂1− , ∂1+ , ◦1 , ε1 ),

(K2 , K1 , ∂2− , ∂2+ , ◦2 , ε2 ),

(K1 , K0 , ∂− , ∂+ , ◦, ε), as partly shown in the diagram K2 ∂1+

∂2− ∂2+

∂1−

K1

/

/ K1 ∂+

∂− ∂+

(4)

∂−

/ / K0

The elements of K0 , K1 , and K2 will be called, respectively, points or objects, edges, and squares. The maps ∂± , ∂i± , i = 1, 2, will be called face maps, the maps εi : 5 The study of partial algebraic operations was initiated in [46]. We can now suggest a reasonable deﬁnition of “higher dimensional algebra” as dealing with families of algebraic operations, whose domains of deﬁnitions are given by geometric conditions.

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K1 −→ K2 , i = 1, 2, resp. ε : K0 −→ K1 will be called degeneracies. The boundaries of an edge and of a square are given by the diagrams ∂1−

/

∂−

/

∂+

∂2−

∂2+ /

1

/2 (5)

∂1+

The partial compositions, ◦1 , resp. ◦2 , are referred to as vertical, resp. horizontal, composition of squares, are deﬁned under the obvious geometric conditions, and have the obvious boundaries. The axioms for a double category also include the usual relations of a 2-cubical set (e.g. ∂− ∂2+ = ∂+ ∂1− ) and the interchange law. We use matrix notation for compositions as a a b = a ◦2 b, = a ◦1 c, c and the crucial interchange law6 for these two compositions allows one to use matrix notation a b a b a b = = c d c d c d for double composites of squares, whenever each row composite and each column composite is deﬁned. We also allow the multiple composition [aij ] of an array (aij ), whenever for all appropriate i, j we have ∂1+ aij = ∂1− ai+1,j , ∂2+ aij = ∂2− ai,j+1 . A clear advantage of double categories and cubical methods is this easy expression of multiple compositions, which allows for algebraic inverse to subdivision, and so applications to local-to-global problems. The identities with respect to ◦1 (vertical identities) are given by ε1 and will be denoted by . Similarly, we have horizontal identities denoted by . Elements of the form ε1 ε(a) = ε2 ε(a) for a ∈ K0 are called double degeneracies and will be denoted by . A morphism of double categories f : K → L consists of a triple of maps fi : Ki → Li , (i = 0, 1, 2), respecting the cubical structure, compositions, and identities. Whereas it is easy to describe a commutative square of morphisms in a category, it is not possible with this amount of structure to describe a commutative cube of squares in a double category. We ﬁrst deﬁne a cube, or 3-shell, that is, without any condition of commutativity, in a double category. 6 The interchange implies that a double monoid is simply an Abelian monoid, so partial algebraic operations are essential for the higher dimensional work.

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Deﬁnition 3.1. Let K be a double category. A cube (3-shell) in K, + − + − + α = (α− 1 , α1 , α2 , α2 , α3 , α3 ),

consists of squares α± i ∈ K2

(i = 1, 2, 3) such that

τ ∂iσ (ατj ) = ∂j−1 (ασi )

for σ, τ = ± and 1 i < j 3.

It is also convenient to have the corresponding notion of square, or 2-shell, of arrows in a category. The obvious compositions also makes these into a double category. It is not hard to deﬁne three compositions of cubes in a double category so that these cubes form a triple category: this is done in [50] or more generally in Section 5 of [51]. A key point is that to deﬁne the notion of a commutative cube, we need extra structure on a double category. Thus, this step up a dimension is nontrivial, as was ﬁrst observed in the groupoid case in [52]. The problem is that a cube has six faces, which easily divide into three even and three odd faces. So we cannot say as we might like that “the cube is commutative if the composition of the even faces equals the composition of the odd faces,” since there are no such valid compositions. The intuitive reason for the need of a new basic structure in that in a 2-dimensional situation, we also need to use the possibility of “turning an edge clockwise or anticlockwise.” The structure to do this is as follows. A connection pair on a double category K is given by a pair of maps

− , + : K1 −→ K2 , whose edges are given by the following diagrams for a ∈ K1 : a/ −

(a) =

a

O O

/ O

O O

O

O 1 = O /o o/ o/ /o O /o /o /o /o O

/2

=

1

1

1 O o/ o/ /o o/ O

+ (a) = 1 O O O / a

a

O o/ o/ /o o/ O = O a O O /

=

/2

1

a

This “hieroglyphic” notation, which was introduced in [53], is useful for expressing the laws these connections satisfy. The ﬁrst is a pair of cancellation laws which read = , = ,

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which can be understood as “if you turn right and then left, you face the same way,” and similarly the other way round. They were introduced in [54]. Note that in this matrix notation, we assume that the edges of the connections are such that the composition is deﬁned. Two other laws relate the connections to the compositions and read = , = . These can be interpreted as “turning left (or right) with your arm outstretched is the same as turning left (or right).” The term “connections” and the name “transport laws” were because these laws were suggested by the laws for path connections in differential geometry, as explained in [47]. It was proved in [49] that a connection pair on a double category K is equivalent to a “thin structure,” namely a morphism of double categories : K1 → K, which is the identity on the edges. The proof requires some “2-dimensional rewriting” using the connections. We can now explain what is a “commutative cube” in a double category K with connection pair. Deﬁnition 3.2. Suppose given, in a double category with connections K, a cube (3-shell) + − + − + α = (α− 1 , α1 , α2 , α2 , α3 , α3 ).

We deﬁne the composition of the odd faces of α to be

α− odd 1 ∂ α= − α3 α+ 2 and the composition of the even faces of α to be

− + α α 2 3 ∂even α = . α+ 1

(6)

(7)

We deﬁne α to be commutative if it satisﬁes the homotopy commutativity lemma (HCL), that is, ∂odd α = ∂even α.

(HCL)

This deﬁnition can be regarded as a cubical, categorical (rather than groupoid) form of the Homotopy Addition Lemma (HAL) in dimension 3. You should draw a 3-shell, label all the edges with letters, and see that this equation makes sense in that the 2-shells of each side of equation (HCL) coincide. Notice, however, that these 2-shells do not have coincident partitions along the edges: that is, the edges of this 2-shell in direction 1 are formed from different compositions of the type 1 ◦ a and a ◦ 1. This deﬁnition is discussed in more detail in [50], is related to other equivalent deﬁnitions, and it is proved that compositions of commutative cubes

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in the three possible directions are also commutative. These results are extended to all dimensions in [55]; this requires the full structure indicated in Section 9 and also the notion of thin element indicated in Section 12. The initial discovery of connections arose in [47] from relating crossed modules to double groupoids. The ﬁrst example of a double groupoid was the double groupoid G of commutative squares in a group G. The ﬁrst step in generalizing this construction c was to consider quadruples a d of elements of G such that abn = cd for some b element n of a subgroup N of G. Experiments quickly showed that for the two compositions of such quadruples to be valid, it was necessary and sufﬁcient that N be normal in G. But in this case, the element n is determined by the boundary, or 2-shell, a, b, c, d. In homotopy theory, we require something more general. Soc we

consider a morphism μ : N → G of groups and consider quintuples n : a d b such that abμ(n) = cd. It then turns out that we get a double groupoid if and only if μ : N → G is a crossed module. The next question is which double groupoids arise in this way? It turns out that we need exactly double groupoids with connection pairs, though in this groupoid case we can deduce − from + using inverses in each dimension. This gives the main result of [47], the equivalence between the category of crossed modules and that of double groupoids with connections and one vertex. These connections were also used in [52] to deﬁne a “commutative cube” in a double groupoid with connections using the equation ⎡ c1 = ⎣−b0

a0−1 c0 a1

⎤ b1 ⎦

representing one face of a cube in terms of the other ﬁve and where the other connections , are obtained from , by using the two inverses in dimension 2. As you might imagine, there are problems in ﬁnding a formula in still higher dimensions. In the groupoid case, this is handled by a homotopy addition lemma and thin elements, [51], but in the category case, a formula for just a commutative four-cube is complicated (see [56]). The blockage of deﬁning a functor to double groupoids was resolved after nine years in 1974 in discussions with Higgins, by considering the Whitehead Theorem W. This showed that a 2-dimensional universal property was available in homotopy theory, which was encouraging; it also suggested that a theory to be any good should recover Theorem W. But this theorem was about relative homotopy groups. This suggested studying a relative situation X∗ : X0 ⊆ X1 ⊆ X. On looking for the simplest way to get a homotopy functor from this situation using squares, the obvious answer came up: consider maps (I 2 , ∂I 2 , ∂∂I 2 ) → (X, X1 , X0 ), that is, maps of the square, which take the edges into X1 and the vertices into X0 , and then take homotopy classes of such maps relative to the vertices of I 2 to form a set ρ2 X∗ . Of course, this set will not inherit a group structure, but the surprise is that it does inherit the structure of double groupoid with connections—the proof is not entirely trivial, and is given in [52] and the expository article [57]. In the case X0 is a singleton, the equivalence

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of such double groupoids to crossed modules takes ρX∗ to the usual second-relative homotopy crossed module. Thus, a search for a higher homotopy groupoid was realized in dimension 2. Connes suggests in [58] that it has been fashionable for mathematicians to disparage groupoids and it might be that a lack of attention to this notion was one reason why such a construction had not been found earlier than 40 years after Hurewicz’s papers. Finding a good homotopy double groupoid led rather quickly, in view of the previous experience, to a substantial account of a 2-dimensional HHvKT [52]. This recovers Theorem W and also leads to new calculations in 2-dimensional homotopy theory, and in fact to some new calculations of 2-types. For a recent summary of some results and some new ones, see the paper published in the J. Symbolic Computation [30]—publication in this journal illustrates that we are interested in using general methods to obtain speciﬁc calculations and ones to which there seems no other route. Once the 2-dimensional case had been completed in 1975, it was easy to conjecture the form of general results for dimensions > 2. These were proved by 1979 and announcements were made in [59] with full details in [51, 60]. However, these results needed a number of new ideas, even just to construct the higher dimensional compositions, and the proof of the HHvKT was quite hard and intricate. Further, for applications such as to explain how the general behaved on homotopies, we also needed a theory of tensor products, found in [61], so that the resulting theory is quite complex. It is also remarkable that ideas of Whitehead in [5] played a key role in these results. 4. Main results Major features of the work over the years with Philip Higgins and others can be summarized in the following diagram of categories and functors:

Diagram 4.1.

in which 4.1.1 the categories FTop of ﬁltered spaces, ω-Gpd of cubical ω-groupoids with connections, and Crs of crossed complexes are monoidal closed, and have a notion of homotopy using ⊗ and a unit interval object; 4.1.2 ρ, are homotopical functors (that is, they are deﬁned in terms of homotopy classes of certain maps) and preserve homotopies;

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4.1.3 λ, γ are inverse adjoint equivalences of monoidal closed categories; 4.1.4 there is a natural equivalence γρ so that either ρ or can be used as appropriate; 4.1.5 ρ, preserve certain colimits and certain tensor products; 4.1.6 the category of chain complexes with a groupoid of operators is monoidal closed, ∇ preserves the monoid structures, and is left adjoint to ; 4.1.7 by deﬁnition, the cubical ﬁltered classifying space is B 2 = | | ◦ U∗ , where U∗ is the forgetful functor to ﬁltered cubical sets7 using the ﬁltration of an ω-groupoid by skeleta, and | | is geometric realization of a cubical set; 4.1.8 there is a natural equivalence ◦ B 2 1; 4.1.9 if C is a crossed complex and its cubical classifying space is deﬁned as B2 C = (B 2 C)∞ , then for a CW -complex X, and using homotopy as in 4.1.1 for crossed complexes, there is a natural bijection of sets of homotopy classes [X, B2 C] ∼ = [X∗ , C]. Recent applications of the simplicial version of the classifying space are in [62–64]. Here, a ﬁltered space consists of a (compactly generated) space X∞ and an increasing sequence of subspaces X∗ : X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ X∞ . With the obvious morphisms, this gives the category FTop. The tensor product in this category is the usual (X∗ ⊗ Y∗ )n = Xp × Yq . p+q=n

The closed structure is easy to construct from the law FTop(X∗ ⊗ Y∗ , Z∗ ) ∼ = FTop(X∗ , FTOP(Y∗ , Z∗ )). An advantage of this monoidal closed structure is that it allows an enrichment of the category FTop over crossed complexes, ω-Gpd using , or ρ applied to FTOP(Y∗ , Z∗ ). The structure of crossed complex is suggested by the canonical example, the fundamental crossed complex X∗ of the ﬁltered space X∗ . So it is given by a diagram 7 Cubical sets are deﬁned, analogously to simplicial sets, as functors K : op → Set, where is the “box” category with objects I n and morphisms the compositions of inclusions of faces and of the various projections I n → I r for n > r. The geometric realization |K| of such a cubical set is obtained by quotienting the disjoint union of the sets K(I n ) × I n by the relations deﬁned by the morphisms of . For more details, see [65], and for variations on the category to include for example connections, see [66]. See also Section 9.

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···

Cn

n

t

C0

Cn−1

···

C2

t

t

C0

C0

2

C1 s

t

C0

Diagram 4.2.

in which in this example, C1 is the fundamental groupoid π1 (X1 , X0 ) of X1 on the “set of base points” C0 = X0 , while for n 2, Cn is the family of relative homotopy groups {Cn (x)} = {πn (Xn , Xn−1 , x) | x ∈ X0 }. The boundary maps are those standard in homotopy theory. There is, for n 2, an action of the groupoid C1 on Cn (and of C1 on the groups C1 (x), x ∈ X0 by conjugation), the boundary morphisms are operator morphisms, δn−1 δn = 0, n 3, and the additional axioms are satisﬁed that 4.3. b−1 cb = cδ2 b , b, c ∈ C2 so that δ2 : C2 → C1 is a crossed module (of groupoids); 4.4. if c ∈ C2 then δ2 c acts trivially on Cn for n 3; 4.5. each group Cn (x) is Abelian for n 3, and so the family Cn is a C1 -module. Clearly, we obtain a category Crs of crossed complexes; this category is not so familiar and so we give arguments for using it in the next section. As algebraic examples of crossed complexes, we have C = C(G, n) where G is a group, commutative if n 2, and C is G in dimension n and trivial elsewhere; C = C(G, 1 : M, n), where G is a group, M is a G-module, n 2, and C is G in dimension 1, M in dimension n, trivial elsewhere, and zero boundary if n = 2; C is a crossed module (of groups) in dimensions 1 and 2 and trivial elsewhere. A crossed complex C has a fundamental groupoid π1 C = C1 / Im δ2 and also, for n 2, a family {Hn (C, p)|p ∈ C0 } of homology groups. 5. Why crossed complexes? • They generalize groupoids and crossed modules to all dimensions. Note that the natural context for second-relative homotopy groups is crossed modules of groupoids rather than groups. • They are good for modelling CW -complexes. • Free crossed resolutions enable calculations with small CW -complexes and CW-maps, see Section 15. • Crossed complexes give a kind of “linear model” of homotopy types, which includes all 2-types. Thus, although they are not the most general model by any means (they do not contain quadratic information such as Whitehead products), this simplicity makes them easier to handle and to relate to classical tools. The new methods and results obtained for crossed complexes can be used as a model

Crossed Complexes and Higher Homotopy Groupoids

• •

•

•

•

101

for more complicated situations. This is how a general n-adic Hurewicz theorem was found [67]. They are convenient for calculation and the functor is classical, involving relative homotopy groups. We explain some results in this form later. They are close to chain complexes with a group(oid) of operators and related to some classical homological algebra (e.g. chains of syzygies). In fact if SX is the simplicial singular complex of a space, with its skeletal ﬁltration, then the crossed complex (SX) can be considered as a slightly noncommutative version of the singular chains of a space. The monoidal structure is suggestive of further developments (e.g. crossed differential algebras), see [68, 69]. It is used in [70] to give an algebraic model of homotopy 3-types and to discuss automorphisms of crossed modules. Crossed complexes have a good homotopy theory, with a cylinder object, and homotopy colimits, [71]. The homotopy classiﬁcation result 4.1.9 generalizes a classical theorem of Eilenberg–Mac Lane. Applications of the simplicial version are given in, for example, [63, 64, 72]. They have an interesting relation with the Moore complex of simplicial groups and of simplicial groupoids (see Section 18).

6. Why cubical ω-groupoids with connections? The deﬁnition of these objects is more difﬁcult to give but will be indicated in Section 9. Here we explain why these structures are a kind of engine giving the power behind the theory. • The functor ρ gives a form of higher homotopy groupoid, thus conﬁrming the visions of the early topologists. • They are equivalent to crossed complexes. • They have a clear monoidal closed structure, and a notion of homotopy, from which one can deduce those on crossed complexes, using the equivalence of categories. • It is easy to relate the functor ρ to tensor products but quite difﬁcult to do this directly for . • Cubical methods, unlike globular or simplicial methods, allow for a simple algebraic inverse to subdivision, which is crucial for our local-to-global theorems. • The additional structure of “connections,” and the equivalence with crossed complexes, allows for the sophisticated notion of commutative cube and the proof that multiple compositions of commutative cubes are commutative. The last fact is a key component of the proof of the HHvKT. • They yield a construction of a (cubical) classifying space B2 C = (B2 C)∞ of a crossed complex C, which generalizes (cubical) versions of Eilenberg–Mac Lane spaces, including the local coefﬁcient case. This has convenient relation to homotopies.

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• There is a current resurgence of the use of cubes in, for example, combinatorics, algebraic topology, and concurrency. There is a Dold–Kan type theorem for cubical Abelian groups with connections [73]. 7. The equivalence of categories Let Crs and ω-Gpd denote, respectively, the categories of crossed complexes and ω-groupoids: we use the latter term as an abbreviation of “cubical ω-groupoids with connections.” A major part of the work consists in deﬁning these categories and proving their equivalence, which, thus, gives an example of two algebraically deﬁned categories whose equivalence is nontrivial. It is even more subtle than that because the functors γ : Crs → ω−Gpd and λ : ω−Gpd → Crs are not hard to deﬁne, and it is easy to prove γλ 1. The hard part is to prove λγ 1, which shows that an ω-groupoid G may be reconstructed from the crossed complex γ (G) it contains. The proof involves using the connections to construct a “folding map” : Gn → Gn , with image γ (G)n , and establishing its major properties, including the relations with the compositions. This gives an algebraic form of some old intuitions of several ways of deﬁning relative homotopy groups, for example, using cubes or cells. On the way, we establish properties of thin elements, as those which fold down to 1, and show that G satisﬁes a strong Kan extension condition, namely that every box has a unique thin ﬁller. This result plays a key role in the proof of the HHvKT for ρ since it is used to show an independence of choice. This part of the proof goes by showing that the two choices can be seen, since we start with a homotopy, as given ± by the two ends ∂n+1 x of an (n + 1)-cube x. It is then shown by induction, using the method of construction and the above result, that x is degenerate in direction n + 1. Hence the two ends in that direction coincide. Properties of the folding map are used also in showing that X∗ is actually included in ρX∗ , in relating two types of thinness for elements of ρX∗ , and in proving a homotopy addition lemma in ρX∗ . Any ω-Gpd G has an underlying cubical set UG. If C is a crossed complex, then the cubical set U(λC) is called the cubical nerve N 2 C of C. It is a conclusion of the theory that we can also obtain N 2 C as (N 2 C)n = Crs(I∗n , C), where I∗n is the usual geometric cube with its standard skeletal ﬁltration. The (cubical) geometric realization |N 2 C| is also called the cubical classifying space B2 C of the crossed complex C. The ﬁltration C∗ of C by skeleta gives a ﬁltration B2 C∗ of B2 C and there is (as in 4.1.6) a natural isomorphism (B2 C∗ ) ∼ = C. Thus, the properties of a crossed complex are those that are universally satisﬁed by X∗ . These proofs use the equivalence of the homotopy categories of Kan8 cubical sets and of 8 The notion of Kan cubical set K is also called a coﬁbrant cubical set. It is an extension condition that any partial r-box in K is the partial boundary of an element of Kr . See, for example, [65], but the idea goes back to the ﬁrst papers by D. Kan [74, 75].

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CW-complexes. We originally took this from the Warwick Masters thesis of S. Hintze, but it is now available with different proofs from Antolini [76] and Jardine [65]. As said above, by taking particular values for C, the classifying space B2 C gives cubical versions of Eilenberg–Mac Lane spaces K(G, n), including the case n = 1 and G noncommutative. If C is essentially a crossed module, then B2 C is called the cubical classifying space of the crossed module, and in fact realizes the k-invariant of the crossed module. Another useful result is that if K is a cubical set, then ρ(|K|∗ ) may be identiﬁed with ρ(K), the free ω-Gpd on the cubical set K, where here |K|∗ is the usual ﬁltration by skeleta. On the other hand, our proof that (|K|∗ ) is the free crossed complex on the nondegenerate cubes of K uses the generalized HHvKT of the next section. It is also possible to give simplicial and globular versions of some of the above results because the category of crossed complexes is equivalent also to those of simplicial T -complexes [77] and of globular ∞-groupoids [78]. In fact, the published paper on the classifying space of a crossed complex [79] is given in simplicial terms to link more easily with well-known theories.

8. First main aim of the work: higher Homotopy van Kampen theorems These theorems give noncommutative tools for higher dimensional local-to-global problems yielding a variety of new, often noncommutative, calculations, which prove (i.e. test) the theory. We now explain these theorems in a way which strengthens the relation with descent, since that was a theme of the conference at which the talk was given on which this survey is based. We suppose given an open cover U = {U λ }λ∈ of X. This cover deﬁnes a map q:E= Uλ → X λ∈

and so we can form an augmented simplicial space ˇ C(q) : · · · E ×X E ×X E

/ / E ×X E / /

/E

q

/ X,

where the higher dimensional terms involve disjoint unions of multiple intersections U ν of the U λ . We now suppose given a ﬁltered space X∗ , a cover U as above of X = X∞ , and ˇ ∗ ), involving multiple intersections U∗ν of an augmented simplicial ﬁltered space C(q the induced ﬁltered spaces. We still need a connectivity condition. Deﬁnition 8.1. A ﬁltered space X∗ is connected if and only if the induced maps π0 X0 → π0 Xn are surjective and πn (Xr , Xn , ν) = 0 for all n > 0, r > n and ν ∈ X0 .

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Theorem 8.2 (Main result (HHvKT)). If U∗ν is connected for all ﬁnite intersections U ν of the elements of the open cover, then (C) (connectivity) X∗ is connected, and ˇ ∗ )), (I) (isomorphism) the following diagram as part of ρ(C(q ρ(E∗ ×X∗ E∗ )

// ρE∗

ρ(q∗ )

/ ρX∗ ,

(cρ)

is a coequalizer diagram. Hence the following diagram of crossed complexes, (E∗ ×X∗ E∗ )

// E∗

(q∗ )

/ X∗ ,

(c)

is also a coequalizer diagram. So we get calculations of the fundamental crossed complex X∗ . It should be emphasized that to get to and apply, this theorem takes just the two papers [51, 60], totalling 58 pages. With this we deduce in the ﬁrst instance • • • • •

the usual vKT for the fundamental groupoid on a set of base points; the Brouwer degree theorem (πn S n = Z); the relative Hurewicz theorem; Whitehead’s theorem that πn (X ∪ {e2λ }, X) is a free crossed module; an excision result, more general than the previous two, on πn (A ∪ B, A, x) as an induced module (crossed module if n = 2) when (A, A∩B) is (n−1)-connected.

The assumptions required of the reader are quite small, just some familiarity with CW complexes. This contrasts with some expositions of basic homotopy theory, where the proof of say the relative Hurewicz theorem requires knowledge of singular homology theory. Of course, it is surprising to get this last theorem without homology, but this is because it is seen as a statement on the morphism of relative homotopy groups πn (X, A, x) → πn (X ∪ CA, CA, x) ∼ = πn (X ∪ CA, x) and is obtained, like our proof of Theorem W, as a special case of an excision result. The reason for this success is that we use algebraic structures that model the underlying processes of the geometry more closely than those in common use. These algebraic structures and their relations are quite intricate, as beﬁts the complications of homotopy theory, so the theory is tight knit. Note also that these results cope well with the action of the fundamental group on higher homotopy groups. The calculational use of the HHvKT for X∗ is enhanced by the relation of with tensor products (see Section 15 for more details). 9. The fundamental cubical ω-groupoid ρX∗ of a ﬁltered space X∗ Here are the basic elements of the construction. I∗n : the n-cube with its skeletal ﬁltration. Set Rn X∗ = FTop(I∗n , X∗ ). This is a cubical set with compositions, connections, and inversions.

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For i = 1, . . . , n, there are standard: face maps ∂i± : Rn X∗ → Rn−1 X∗ ; degeneracy maps εi : Rn−1 X∗ → Rn X∗ connections i± : Rn−1 X∗ → Rn X∗ compositions a ◦i b deﬁned for a, b ∈ Rn X∗ such that ∂i+ a = ∂i− b inversions −i : Rn → Rn . The connections are induced by γiα : I n → I n−1 deﬁned using the monoid structures max, min : I 2 → I. They are essential for many reasons, for example, to discuss the notion of commutative cube. These operations have certain algebraic properties that are easily derived from the geometry and which we do not itemize here (see, e.g. [80]). These were listed ﬁrst in the Bangor thesis of Al-Agl [81] (In the paper [51], the only basic connections − needed are the + i , from which the i are derived using the inverses of the groupoid structures.). Now it is natural and convenient to deﬁne f ≡ g for f, g : I∗n → X∗ to mean f is homotopic to g through ﬁltered maps and relative to the vertices of I n . This gives a quotient map p : Rn X∗ → ρn X∗ = (Rn X∗ / ≡). The following results are proved in [60]. 9.1. The compositions on RX∗ are inherited by ρX∗ to give ρX∗ , the structure of cubical multiple groupoid with connections. 9.2. The map p : RX∗ → ρX∗ is a Kan ﬁbration of cubical sets. The proofs of both results use methods of collapsing, which are indicated in the next section. The second result is almost unbelievable. Its proof has to give a systematic method of deforming a cube with the right faces “up to homotopy” into a cube with exactly the right faces, using the given homotopies. In both cases, the assumption that the relation ≡ uses homotopies relative to the vertices is essential to start the induction. (In fact, the paper [60] does not use homotopy relative to the vertices, but imposes an extra condition J0 , that each loop in X0 is contractible X1 , which again starts the induction. This condition is awkward in applications, for example, to function spaces. A full exposition of the whole story is in preparation [82].) An essential ingredient in the proof of the HHvKT is the notion of multiple composition. We have discussed this already in dimension 2, with a suggestive picture in the diagram (3). In dimension n, the aim is to give algebraic expression to the idea of a cube I n being subdivided by hyperplanes parallel to the faces into many small cubes, a subdivision with a long history in mathematics. Let (m) = (m1 , . . . , mn ) be an n-tuple of positive integers and φ(m) : I n → [0, m1 ] × · · · × [0, mn ] be the map (x1 , . . . , xn ) → (m1 x1 , . . . , mn xn ). Then a subdivision of type (m) of a map α : I n → X is a factorization α = α ◦ φ(m) ; its parts are the cubes α(r) , where (r) = (r1 , . . . , rn ) is an n-tuple of integers with 1 ri mi , i = 1, . . . , n, and

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where α(r) : I n → X is given by (x1 , . . . , xn ) → α (x1 + r1 − 1, . . . , xn + rn − 1). We then say that α is the composite of the cubes α(r) and write α = [α(r) ]. The domain of α(r) is then the set {(x1 , . . . , xn ) ∈ I n : ri − 1 xi ri , 1 i n}. This ability to express “algebraic inverse to subdivision” is one beneﬁt of using cubical methods. Similarly, in a cubical set with compositions satisfying the interchange law, we can deﬁne the multiple composition [α(r) ] of a multiple array (α(r) ) provided the obviously necessary multiple incidence relations of the individual α(r) to their neighbors are satisﬁed. Here is an application that is essential in many proofs and seems hard to prove without the techniques involved in 9.2. Theorem 9.3 (Lifting multiple compositions). Let [α(r) ] be a multiple composition in ρn X∗ . Then representatives a(r) of the α(r) may be chosen so that the multiple composition [a(r) ] is well deﬁned in Rn X∗ . Proof: The multiple composition [α(r) ] determines a cubical map A : K → ρX∗ , where the cubical set K corresponds to a representation of the multiple composition by a subdivision of the geometric cube so that top cells c(r) of K are mapped by A to α(r) . Consider the diagram, in which ∗ is a corner vertex of K, / RX∗ . ∗ |> | | A | p | | | | / ρX∗ K A

Then K collapses to ∗, written K ∗. (As an example, see how the subdivision in the diagram (3) may be collapsed row by row to a point.) By the ﬁbration result, A lifts to A , which represents [a(r) ], as required. So we have to explain collapsing. 10. Collapsing We use a basic notion of collapsing and expanding due to J.H.C. Whitehead [6]. Let C ⊆ B be subcomplexes of I n . We say C is an elementary collapse of B, B e C, if for some s 1, there is an s-cell a of B and (s − 1)-face b of a, the free face, such that B = C ∪ a,

C ∩ a = a˙ \ b

(where a˙ denotes the union of the proper faces of a).

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We say B1 collapses to Br , written B1 Br , if there is a sequence B1 e B2 e · · · e Br of elementary collapses. If C is a subcomplex of B, then B × I (B × {0} ∪ C × I ) (this is proved by induction on dimension of B \ C). Further, I n collapses to any one of its vertices (this may be proved by induction on n using the ﬁrst example). These collapsing techniques allow the construction of the extensions of ﬁltered maps and ﬁltered homotopies that are crucial for proving 9.1 that ρX∗ does obtain the structure of multiple groupoid. However, more subtle collapsing techniques using partial boxes are required to prove the ﬁbration theorem 9.2, as partly explained in the next section. 11. Partial boxes Let C be an r-cell in the n-cube I n . Two (r − 1)-faces of C are called opposite if they do not meet. A partial box in C is a subcomplex B of C generated by one (r − 1)-face b of C (called a base of B) and a number, possibly zero, of other (r − 1)-faces of C, none of which is opposite to b. The partial box is a box if its (r − 1)-cells consist of all but one of the (r − 1)-faces of C. The proof of the ﬁbration theorem uses a ﬁlter homotopy extension property and the following: Proposition 11.1 (Key Proposition). Let B, B be partial boxes in an r-cell C of I n such that B ⊆ B. Then there is a chain B = Bs Bs−1 · · · B1 = B such that (i) each Bi is a partial box in C; (ii) Bi+1 = Bi ∪ ai , where ai is an (r − 1)-cell of C not in Bi ; (iii) ai ∩ Bi is a partial box in ai . The proof is quite neat and follows the pictures. Induction up such a chain of partial boxes is one of the steps in the proof of the ﬁbration theorem 9.2. The proposition implies that an inclusion of partial boxes is what is known as an anodyne extension [65]. Here is an example of a sequence of collapsings of a partial box B, which illustrate some choices in forming a collapse B 0 through two other partial boxes B1 and B2 .

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e

e

e

e B1

B

e

e

e

B2

The proof of the ﬁbration theorem gives a program for carrying out the deformations needed to do the lifting. In some sense, it implies computing a multiple composition that can be done using collapsing as the guide. Methods of collapsing generalize methods of trees in dimension 1. 12. Thin elements Another key concept is that of thin element α ∈ ρn X∗ for n 2. The proofs here use strong results of [51]. We say α is geometrically thin if it has a deﬁcient representative, that is, an a : I∗n → X∗ such that a(I n ) ⊆ Xn−1 . We say α is algebraically thin if it is a multiple composition of degenerate elements or those coming from repeated (including 0) negatives of connections. Clearly, any multiple composition of algebraically thin elements is thin. Theorem 12.1. (i) Algebraically thin is equivalent to geometrically thin. (ii) In a cubical ω-groupoid with connections, any box has a unique thin ﬁller. Proof. The proof of the forward implication in (i) uses lifting of multiple compositions, in a stronger form than stated above. The proofs of (ii) and the backward implication in (i) use the full force of the algebraic relation between ω-groupoids and crossed complexes. These results allow one to replace arguments with commutative cubes by arguments with thin elements. 13. Sketch proof of the HHvKT The proof goes by verifying the required universal property. Let U be an open cover of X as in Theorem 8.2. ˇ ∗ )), We go back to the following diagram, whose top row is part of ρ(C(q ρ(E∗ ×X∗ E∗ )

∂0 ∂1

ρ(q∗ ) / ρ(E ) / ρX∗ / ∗ K KK KK KK KK f K KK f KK K% G

(cρ)

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To prove this top row is a coequalizer diagram, we suppose given a morphism f : ρ(E∗ ) → G of cubical ω-groupoids with connection such that f ◦ ∂0 = f ◦ ∂1 , and we prove that there is a unique morphism f : ρX∗ → G such that f ◦ ρ(q∗ ) = f . To deﬁne f (α) for α ∈ ρX∗ , you subdivide a representative a of α to give a = [a(r) ] so that each a(r) lies in an element U (r) of U; use the connectivity conditions and this subdivision to deform a into b = [b(r) ] so that b(r) ∈ R(U∗(r) ) and so obtain β(r) ∈ ρ(U∗(r) ). The elements fβ(r) ∈ G may be composed in G (by the conditions on f ) to give an element θ(α) = [fβ(r) ] ∈ G. So the proof of the universal property has to use an algebraic inverse to subdivision. Again an analogy here is with sending an email: the element you start with is subdivided, deformed so that each part is correctly labelled, the separate parts are sent, and then recombined. The proof that θ(α) is independent of the choices made uses crucially properties of thin elements. The key point is a ﬁlter homotopy h : α ≡ α in Rn X∗ gives a deﬁcient element of Rn+1 X∗ . The method is to do the subdivision and deformation argument on such a homotopy, push the little bits in some ρn+1 (U∗λ ) (now thin) over to G, combine them, and get a thin element τ ∈ Gn+1 , all of whose faces not involving the direction (n + 1) are thin because h was given to be a ﬁlter homotopy. An inductive argument on unique thin ﬁllers of boxes then shows that τ is degenerate in direction (n + 1) so that the two ends in direction (n + 1) are the same. This ends a rough sketch of the proof of the HHvKT for ρ. Note that the theory of these forms of multiple groupoids is designed to make this last argument work. We replace a formula for saying a cube h has commutative boundary by a statement that h is thin. It would be very difﬁcult to replace the above argument, on the composition of thin elements, by a higher dimensional manipulation of formulae such as that given in Section 3 for a commutative 3-cube. Further, the proof does not require knowledge of the existence of all coequalizers, nor does it give a recipe for constructing these in speciﬁc examples.

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14. Tensor products and homotopies The construction of the monoidal closed structure on the category ω-Gpd is based on rather formal properties of cubical sets, and the fact that for the cubical set In , we have Im ⊗ In ∼ = Im+n . The details are given in [61]. The equivalence of categories implies then that the category Crs is also monoidal closed, with a natural isomorphism Crs(A ⊗ B, C) ∼ = Crs(A, CRS(B, C)). Here the ‘internal hom’ CRS(B, C) is a crossed complex and its elements in dimension n are: for n = 0 the morphisms B → C; for n = 1 are homotopies of morphisms; and for n 2 are forms of higher homotopies. The precise description of these is obtained of course by tracing out in detail the equivalence of categories. It should be emphasized that certain choices are made in constructing this equivalence, and these choices are reﬂected in the ﬁnal formulae that are obtained. An important result is that if X∗ , Y∗ are ﬁltered spaces, then there is a natural transformation η : ρX∗ ⊗ ρY∗ → ρ(X∗ ⊗ Y∗ ) [a] ⊗ [b] → [a ⊗ b], where if a : I∗m → X∗ , b : I∗n → Y∗ , then a ⊗ b : I∗m+n → X∗ ⊗ Y∗ . It is not hard to see, in this cubical setting, that η is well deﬁned. It can also be shown using previous results that η is an isomorphism if X∗ , Y∗ are the geometric realizations of cubical sets with the usual skeletal ﬁltration. The equivalence of categories now gives a natural transformation of crossed complexes η : X∗ ⊗ Y∗ → (X∗ ⊗ Y∗ ).

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It would be hard to construct this directly. It is proved in [79] that η is an isomorphism if X∗ , Y∗ are the skeletal ﬁltrations of CW -complexes. The proof uses the HHvKT, and the fact that A ⊗ − on crossed complexes has a right adjoint and so preserves colimits. It is proved in [69] that η is an isomorphism if X∗ , Y∗ are coﬁbred, connected ﬁltered spaces. This applies, in particular, to the useful case of the ﬁltration B2 C∗ of the classifying space of a crossed complex. It turns out that the deﬁning rules for the tensor product of crossed complexes, which follows from the above construction, are obtained as follows. We ﬁrst deﬁne a bimorphism of crossed complexes. Deﬁnition 14.1. A bimorphism θ : (A, B) → C of crossed complexes is a family of maps θ : Am × Bn → Cm+n satisfying the following conditions, where a ∈ Am , b ∈ Bn , a1 ∈ A1 , b1 ∈ B1 (temporarily using additive notation throughout the deﬁnition): (i) β(θ(a, b)) = θ(βa, βb) for all a ∈ A, b ∈ B.

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(ii) θ(a, bb1 ) = θ(a, b)θ(βa,b1 ) if m 0, n 2, θ(aa1 , b) = θ(a, b)θ(a1 ,βb) if m 2, n 0. (iii) θ(a, b + b ) θ(a, b) + θ(a, b ) if m = 0, n 1 or m 1, n 2, = ) θ(βa,b θ(a, b) + θ(a, b ) if m 1, n = 1, θ(a + a , b) θ(a, b) + θ(a , b) = θ(a , b) + θ(a, b)θ(a ,βb)

if m 1, n = 0 or m 2, n 1, if m = 1, n 1.

(iv)

δm+n (θ(a, b)) =

⎧ ⎪ θ(δm a, b) + (−)m θ(a, δn b) ⎪ ⎪ ⎪ ⎪ ⎨ −θ(a, δn b) − θ(βa, b) + θ(αa, b)θ(a,βb)

if m 2, n 2, if m = 1, n 2,

⎪ (−)m+1 θ(a, βb) + (−)m θ(a, αb)θ(βa,b) + θ(δm a, b) if m 2, n = 1, ⎪ ⎪ ⎪ ⎪ ⎩ −θ(βa, b) − θ(a, αb) + θ(αa, b) + θ(a, βb) if m = n = 1.

(v)

δm+n (θ(a, b)) =

θ(a, δn b)

if m = 0, n 2 ,

θ(δm a, b)

if m 2, n = 0 .

(vi) α(θ(a, b)) = θ(a, αb)

and β(θ(a, b)) = θ(a, βb) if m = 0, n = 1 ,

α(θ(a, b)) = θ(αa, b)

and

β(θ(a, b)) = θ(βa, b) if m = 1, n = 0 .

The tensor product of crossed complexes A, B is given by the universal bimorphism (A, B) → A ⊗ B, (a, b) → a ⊗ b. The rules for the tensor product are obtained by replacing θ(a, b) by a ⊗ b in the above formulae. The conventions for these formulae for the tensor product arise from the derivation of the tensor product via the category of cubical ω-groupoids with connections, and the formulae are forced by our conventions for the equivalence of the two categories [51, 61]. The complexity of these formulae is directly related to the complexities of the cell structure of the product Em × En , where the n-cell En has cell structure e0 if n = 0, e0± ∪ e1 if n = 1, and e0 ∪ en−1 ∪ en if n 2.

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It is proved in [61] that the bifunctor − ⊗ − is symmetric and that if a0 is a vertex of A, then the morphism B → A ⊗ B, b → a0 ⊗ b is injective. There is a standard groupoid model I of the unit interval, namely the indiscrete groupoid on two objects 0, 1. This is easily extended trivially to either a crossed complex or an ω-Gpd. So using ⊗, we can deﬁne a “cylinder object” I ⊗ − in these categories and so a homotopy theory [71]. 15. Free crossed complexes and free crossed resolutions Let C be a crossed complex. A free basis B∗ for C consists of the following: B0 is set which we take to be C0 ; B1 is a graph with source and target maps s, t : B1 → B0 and C1 is the free groupoid on the graph B1 : that is, B1 is a subgraph of C1 and any graph morphism B1 → G to a groupoid G extends uniquely to a groupoid morphism C1 → G; Bn is, for n 2, a totally disconnected subgraph of Cn with target map t : Bn → B0 ; for n = 2, C2 is the free crossed C1 -module on B2 while for n > 2, Cn is the free (π1 C)-module on Bn . It may be proved using the HHvKT that if X∗ is a CW -complex with the skeletal ﬁltration, then X∗ is the free crossed complex on the characteristic maps of the cells of X∗ . It is proved in [79] that the tensor product of free crossed complexes is free. A free crossed resolution F∗ of a groupoid G is a free crossed complex that is aspherical together with an isomorphism φ : π1 (F∗ ) → G. Analogs of standard methods of homological algebra show that free crossed resolutions of a group are unique up to homotopy equivalence. To apply this result to free crossed resolutions, we need to replace free crossed resolutions by CW-complexes. A fundamental result for this is the following, which goes back to Whitehead [6] and Wall [83], and which is discussed further by Baues in [84, Chapter VI, Section 7]: Theorem 15.1. Let X∗ be a CW -ﬁltered space, and let φ : X∗ → C be a homotopy equivalence to a free crossed complex with a preferred free basis. Then, there is a CW-ﬁltered space Y∗ , and an isomorphism Y∗ ∼ = C of crossed complexes with preferred basis such that φ is realized by a homotopy equivalence X∗ → Y∗ . In fact, as pointed out by Baues, Wall states his result in terms of chain complexes, but the crossed complex formulation seems more natural, and avoids questions of realizability in dimension 2, which are unsolved for chain complexes. Corollary 15.2. If A is a free crossed resolution of a group G, then A is realized as free crossed complex with preferred basis by some CW -ﬁltered space Y∗ . Proof. We only have to note that the group G has a classifying CW -space BG, whose fundamental crossed complex (BG) is homotopy equivalent to A.

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Baues also points out in [84, p. 657] an extension of these results, which we can apply to the realization of morphisms of free crossed resolutions. A new proof of this extension is given by Faria Martins in [85], using methods of Ashley [77]. Proposition 15.3. Let X = K(G, 1), Y = K(H, 1) be CW -models of Eilenberg– Mac Lane spaces and let h : X∗ → (Y∗ ) be a morphism of their fundamental crossed complexes with the preferred bases given by skeletal ﬁltrations. Then h = (g) for some cellular g : X → Y . Proof. Certainly, h is homotopic to (f ) for some f : X → Y since the set of pointed homotopy classes X → Y is bijective with the morphisms of groups A → B. The result follows from [84, p. 657, (**)] (if f is -realizable, then each element in the homotopy class of f is -realizable). These results are exploited in [86, 87] to calculate free crossed resolutions of the fundamental groupoid of a graph of groups. An algorithmic approach to the calculation of free crossed resolutions for groups is given in [17] by constructing partial contracting homotopies for the universal cover at the same time as constructing this universal cover inductively. This has been implemented in GAP4 by Heyworth and Wensley [88].

16. Classifying spaces and the homotopy classiﬁcation of maps The formal relations of cubical sets and of cubical ω-groupoids with connections and the relation of Kan cubical sets with topological spaces allow the proof of a homotopy classiﬁcation theorem. Theorem 16.1. If K is a cubical set and G is an ω-groupoid, then there is a natural bijection of sets of homotopy classes [|K|, |UG|] ∼ = [ρ(|K|∗ ), G], where on the left-hand side, we work in the category of spaces, and on the right in ω-groupoids. Here, |K|∗ is the ﬁltration by skeleta of the geometric realization of the cubical set. We explained earlier how to deﬁne a cubical classifying space say B2 (C) of a crossed complex C as B2 (C) = |UN 2 C| = |UλC|. The properties already stated now give the homotopy classiﬁcation theorem 4.1.9. It is shown in [60] that for a CW -complex Y , there is a map p : Y → B2 Y∗ , whose homotopy ﬁbre is n-connected if Y is connected and πi Y = 0 for 2 i n−1. It follows that if also X is a connected CW -complex with dim X n, then p induces a bijection [X, Y ] → [X, BY∗ ].

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So, under these circumstances, we get a bijection [X, Y ] → [X∗ , Y∗ ].

(9)

This result, due to Whitehead [5], translates a topological homotopy classiﬁcation problem to an algebraic one. We explain below how this result can be translated to a result on chain complexes with operators. It is also possible to deﬁne a simplicial nerve N (C) of a crossed complex C by N (C)n = Crs((n ), C). The simplicial classifying space of C is then deﬁned using the simplicial geometric realization B (C) = |N (C)|. The properties of this simplicial classifying space are developed in [79], and in particular an analog of 4.1.9 is proved. The simplicial nerve and an adjointness Crs((L), C) ∼ = Simp(L, N (C)) are used in [89, 90] for an equivariant homotopy theory of crossed complexes and their classifying spaces. Important ingredients in this are notions of coherence and an Eilenberg–Zilber type theorem for crossed complexes proved in Tonks’ Bangor thesis [91, 92], see also [93]. Labesse in [94] deﬁnes a crossed set. In fact, a crossed set is exactly a crossed module (of groupoids) δ : C → X G, where G is a group acting on the set X and X G is the associated actor groupoid; thus, the simplicial construction from a crossed set described by Larry Breen in [94] is exactly the simplicial nerve of the crossed module, which is regarded as a crossed complex. Hence, the cohomology with coefﬁcients in a crossed set used in [94] is a special case of cohomology with coefﬁcients in a crossed complex, dealt with in [79] (We are grateful to Breen for pointing this out to us in 1999.).

17. Relation with chain complexes with a groupoid of operators Chain complexes with a group of operators are a well-known tool in algebraic topol∗ of cellular chains of the ogy, where they arise naturally as the chain complex C∗ X universal cover X∗ of a reduced CW -complex X∗ . The group of operators here is the fundamental group of the space X. Whitehead in [5] gave an interesting relation between his free crossed complexes (he called them “homotopy systems”) and such chain complexes. We refer later to his important homotopy classiﬁcation results in this area. Here, we explain the relation with the Fox free differential calculus [95].

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Let μ : M → P be a crossed module of groups and let G = Cokerμ. Then, there is an associated diagram μ

M h2

φ

/P h1

M ab

∂2

/D φ

/G

(10)

h0

∂1

/ Z[G]

in which the second row consists of (right) G-modules and module morphisms. Here, h2 is simply the Abelianization map; h1 : P → Dφ is the universal φ-derivation, that is, it satisﬁes h1 (pq) = h1 (p)φq + h1 (q), for all p, q ∈ P, and is universal for this property; and h0 is the usual derivation g → g − 1. Whitehead in his Lemma 7 of [5] gives this diagram in the case P is a free group, when he takes Dφ to be the free G-module on the same generators as the free generators of P. Our formulation, which uses the derived module due to Crowell [96], includes his case. It is remarkable that diagram (10) is a commutative diagram in which the vertical maps are operator morphisms and that the bottom row is deﬁned by this property. The proof in [97] follows essentially Whitehead’s proof. The bottom row is exact: this follows from results in [96], and is a reﬂection of a classical fact on group cohomology, namely the relation between central extensions and the Ext functor, (see [21]). In the case the crossed module is the crossed module δ : C(ω) → F(X) derived from a presentation of a group, then C(ω)ab is isomorphic to the free G-module on R, Dφ is the free G-module on X, and it is immediate from the above that ∂2 is the usual derivative (∂r/∂x) of Fox’s free differential calculus [95]. Thus, Whitehead’s results anticipate those of Fox. It is also proved in [5] that if the restriction M → μ(M) of μ has a section, which is a morphism but not necessarily a P-map, then h2 maps Ker μ isomorphically to Ker ∂2 . This allows calculation of the module of identities among relations by using module methods, and this is commonly exploited, see for example [16] and the references there. Whitehead introduced the categories CW of reduced CW -complexes, HS of homotopy systems, and FCC of free chain complexes with a group of operators, together with functors

C

CW −→ HS −→ FCC. In each of these categories, he introduced notions of homotopy and he proved that C induces an equivalence of the homotopy category of HS with a subcategory of the homotopy category of FCC. Further, CX∗ is isomorphic to the chain complex ∗ of cellular chains of the universal cover of X so that under these circumstances, C∗ X there is a bijection of sets of homotopy classes ∗ , C∗ Y∗ ]. [X∗ , Y∗ ] → [C∗ X

(11)

This with the bijection (9) can be interpreted as an operator version of the Hopf classiﬁcation theorem. It is surprisingly little known. It includes results of Olum [98]

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published later, and it enables quite useful calculations to be done easily, such as the homotopy classiﬁcation of maps from a surface to the projective plane [99], and other cases. Thus, we see once again that this general theory leads to speciﬁc calculations. All these results are generalized in [97] to the nonfree case and to the nonreduced case, which requires a groupoid of operators, thus giving functors

∇

FTop −→ Crs −→ Chain. (The paper [97] uses the notation for this ∇.) One utility of the generalization to groupoids is that the functor ∇ then has a right adjoint and so preserves colimits. An example of this preservation is given in [97, Example 2.10]. The construction of the right adjoint to ∇ builds on a number of constructions used earlier in homological algebra. The deﬁnitions of the categories under consideration to obtain a generalization of the bijection (11) has to be quite careful, since it works in the groupoid case, and not all morphisms of the chain complex are realizable. This analysis of the relations between these two categories is used in [79] to give an account of cohomology with local coefﬁcients. It is also proved in [97] that the functor ∇ preserves tensor products, where the tensor in the category Chain is a generalization to modules over groupoids of the usual tensor for chain complexes of modules of groups. Since the tensor product is described explicitly in dimensions 2 in [61] and (∇C)n = Cn for n 3, this preservation yields a complete description of the tensor product of crossed complexes.

18. Crossed complexes and simplicial groups and groupoids The Moore complex NG of a simplicial group G is not in general a (reduced) crossed complex. Let Dn G be the subgroup of Gn generated by degenerate elements. Ashley showed in his thesis [77] that NG is a crossed complex if and only if (NG)n ∩(DG)n = {1} for all n 1. Ehlers and Porter in [100, 101] show that there is a functor C from simplicial groupoids to crossed complexes in which C(G)n is obtained from N(G)n by factoring out (NGn ∩ Dn )dn+1 (NGn+1 ∩ Dn+1 ), where the Moore complex is deﬁned so that its differential comes from the last simplicial face operator. This is one part of an investigation into the Moore complex of a simplicial group, of which the most general investigation is by Carrasco and Cegarra in [102]. An important observation in [103] is that if N G is an inclusion of a normal simplicial subgroup of a simplicial group, then the induced morphism on components π0 (N) → π0 (G) obtains the structure of crossed module. This is directly analogous to the fact that if F → E → B is a ﬁbration sequence, then the induced morphism of fundamental groups π1 (F, x) → π1 (E, x) also obtains the structure of crossed

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module. This last fact is relevant to algebraic K-theory, where for a ring R, the homotopy ﬁbration sequence is taken to be F → B(GL(R)) → B(GL(R))+ .

19. Other homotopy multiple groupoids A natural question is whether there are other useful forms of higher homotopy groupoids. It is because the geometry of convex sets is so much more complicated in dimensions > 1 than in dimension 1 that new complications emerge for the theories of higher order group theory and of higher homotopy groupoids. We have different geometries, for example, those of disks, globes, simplices, cubes, as shown in dimension 2 in the following diagram.

The cellular decomposition for an n-disk is Dn = e0 ∪ en−1 ∪ en and that for globes is n Gn = e0± ∪ e1± ∪ · · · ∪ en−1 ± ∪e .

The higher dimensional group(oid) theory reﬂecting the n-disks is that of crossed complexes and that for the n-globes is called globular ω-groupoids. A common notion of higher dimensional category is that of n-category, which generalize the 2-categories studied in the late 1960s. A 2-category C is a category enriched in categories, in the sense that each hom set C(x, y) is given the structure of category, and there are appropriate axioms. This gives inductively the notion of an n-category as a category enriched in (n − 1)-categories. This is called a “globular” approach to higher categories. The notion of n-category for all n was axiomatized in [78] and called an ∞-category; its underlying geometry was also axiomatized in this paper as a family of sets Sn , n 0 with operations Diα : Sn → Si , Ei : Si → Sn , α = 0, 1; i = 1, . . . , n − 1 satisfying appropriate laws. Such a structure was later called a “globular set” [104], and the term ω-category was used instead of the earlier ∞-category. Difﬁculties of the globular approach are to deﬁne multiple compositions, and also monoidal closed structures, although these are clear in the cubical approach. A globular higher homotopy groupoid of a ﬁltered space has been constructed in [105], deduced from cubical results in [60, 80]. Although the proof of the HHvKT outlined earlier does seem to require cubical methods, there is still a question of the place of globular and simplicial methods in this area. A simplicial analog of the equivalence of categories is given in [77, 106], using Dakin’s notion of simplicial T -complex [107]. However, it is difﬁcult to describe in detail the notion of tensor product of such structures or to formulate a proof of the

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HHvKT theorem in that context. There is a tendency to replace the term T -complex from all this earlier work such as [77, 108] by complicial set [109]. It is easy to deﬁne a homotopy globular set ρ X∗ of a ﬁltered space X∗ , but it is not quite so clear how to prove directly that the expected compositions are well deﬁned. However, there is a natural graded map i : ρ X∗ → ρX∗

(12)

and applying the folding map of [80,81] analogously to methods in [60] allows one to prove that i of (12) is injective. It follows that the compositions on ρX∗ are inherited by ρ X∗ to make the latter a globular ω-groupoid. The details are in [105]. Loday in 1982 [110] deﬁned the fundamental catn -group of an n-cube of spaces (a catn -group may be deﬁned as an n-fold category internal to the category of groups) and showed that catn -groups model all reduced weak homotopy (n + 1)-types. Joint work [7] formulated and proved a HHvKT for the catn -group functor from n-cubes of spaces. This allows new local to global calculations of certain homotopy n-types [111] and also an n-adic Hurewicz theorem [67]. This work obtains more powerful results than the purely linear theory of crossed complexes. It yields a group-theoretic description of the ﬁrst nonvanishing homotopy group of a certain (n+1)-ad of spaces, so several formulae for the homotopy and homology groups of speciﬁc spaces; [112] gives new applications. Porter in [103] gives an interpretation of Loday’s results using methods of simplicial groups. There is clearly a lot to do in this area. See [113] for relations of catn -groups with homological algebra. Recently, some absolute homotopy 2-groupoids and double groupoids have been deﬁned, see [114] and the references there, while [115] applies generalized Galois theory to give a new homotopy double groupoid of a map, generalizing previous work of [52]. It is signiﬁcant that crossed modules have been used in a differential topology situation by Mackaay and Picken [116]. Reinterpretations of these ideas in terms of double groupoids are started in [117]. It seems reasonable to suggest that in the most general case, double groupoids are still somewhat mysterious objects. The paper [118] gives a kind of classiﬁcation of them. 20. Conclusion and questions • The emphasis on ﬁltered spaces rather than the absolute case is open to question. But no useful deﬁnition of a higher homotopy groupoid of a space with clear uses has been proposed. The work of Loday makes use of n-cubes of ﬁltered spaces (see [110, 119]). • Mirroring the geometry by the algebra is crucial for conjecturing and proving universal properties. • Thin elements are crucial for modelling a concept not so easy to deﬁne or handle algebraically, that of commutative cubes. (see also [55, 120]). • The cubical methods summarized in Section 9 have also been applied in concurrency theory, (see, e.g. [121, 122].)

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• HHvKT theorems give, when they apply, exact information even in noncommutative situations. The implications of this for homological algebra could be important. • One construction inspired eventually by this work, the non-Abelian tensor product of groups, has a bibliography of 90 papers, since it was deﬁned with Loday in [7]. • Globular methods do ﬁt into this scheme. They have not so far yielded new calculations in homotopy theory (see [105]) but have been applied to directed homotopy theory [121]. Globular methods are the main tool in approaches to weak category theory (see, e.g. [104, 123]) although the potential of cubical methods in that area is hinted at in [120]. • For computations, we really need strict structures (although we do want to compute invariants of homotopy colimits). • No work seems to have been done on Poincaré duality, that is, on ﬁnding special qualities of the fundamental crossed complex of the skeletal ﬁltration of a combinatorial manifold. However, the book by Sharko ([124, Chapter VI]) does use crossed complexes for investigating Morse functions on a manifold. • In homotopy theory, identiﬁcations in low dimensions can affect highdimensional homotopy. So we need structure in a range of dimensions to model homotopical identiﬁcations algebraically. The idea of identiﬁcations in low dimensions is reﬂected in the algebra by “induced constructions.” • In this way, we calculate some crossed modules modeling homotopy 2-types, whereas the corresponding k-invariant is often difﬁcult to calculate. ˇ • The use of crossed complexes in Cech theory is a current project with Jim Glazebrook and Tim Porter. • Question: Are there applications of higher homotopy groupoids in other contexts where the fundamental groupoid is currently used, such as algebraic geometry? • Question: There are uses of double groupoids in differential geometry, for example in Poisson geometry, and in 2-dimensional holonomy [125]. Is there a nonAbelian De Rham theory, using an analog of crossed complexes? • Question: Is there a truly noncommutative integration theory based on limits of multiple compositions of elements of multiple groupoids, in analogy to the way length is deﬁned using limits of sequences of line segments?

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[61] R. Brown, P.J. Higgins, Tensor products and homotopies for ω-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987), 1–33. [62] R. Brown, Exact sequences of ﬁbrations of crossed complexes, homotopy classiﬁcation of maps, and nonabelian extensions of groups, J. Homotopy and Related Structures 3 (2008), 331–343. [63] T. Porter, V. Turaev, Formal Homotopy Quantum Field Theories I: Formal maps and crossed C-algebras, J. Homotopy and Related Structures, 3 (2008), 113–159. [64] J. Faria Martins, T. Porter, On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups, Theory Appl. Categories 18 (2007), 118–150. [65] J.F. Jardine, Categorical homotopy theory, Homology, Homotopy Appl. 8 (2006), 71–144. [66] M. Grandis, L. Mauri, Cubical sets and their site, Theory Applic. Categories 11 (2003), 185–201. [67] R. Brown, J.-L. Loday, Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc. (3) 54 (1987), 176–192. [68] H.-J. Baues, A. Tonks, On the twisted cobar construction, Math. Proc. Cambridge Philos. Soc. 121 (1997), 229–245. [69] H.J. Baues, R. Brown, On relative homotopy groups of the product ﬁltration, the James construction, and a formula of Hopf, J. Pure Appl. Algebra 89 (1993), 49–61. [70] R. Brown, N.D. Gilbert, Algebraic models of 3-types and automorphism structures for crossed modules, Proc. London Math. Soc. (3) 59 (1989), 51–73. [71] R. Brown, M. Golasi´nski, A model structure for the homotopy theory of crossed complexes, Cah. Top. Géom. Diff. Cat. 30 (1989), 61–82. [72] J. Faria Martins, On the homotopy type and fundamental crossed complex of the skeletal ﬁltration of a CW-complex, Homology, Homotopy and Applications 9 (2007), 295–329. [73] R. Brown, P.J. Higgins, Cubical Abelian groups with connections are equivalent to chain complexes, Homology, Homotopy and Applications 5 (2003), 49–52. [74] D.M. Kan, Abstract homotopy. I. Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 1092–1096. [75] D.M. Kan, Abstract homotopy. II. Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 255–258. [76] R. Antolini, Cubical Structures and Homotopy Theory, Ph.D. thesis, Univ. Warwick, Coventry, 1996. [77] N. Ashley, Simplicial T-Complexes: a non Abelian version of a theorem of Dold-Kan, Dissertationes Math. 165 (1988), 11–58, (Published version of University of Wales PhD thesis, 1978). [78] R. Brown, P.J. Higgins, The equivalence of ∞-groupoids and crossed complexes, Cahiers Top. Géom. Diff. 22 (1981), 370–386. [79] R. Brown, P.J. Higgins, The classifying space of a crossed complex, Math. Proc. Cambridge Philos. Soc. 110 (1991), 95–120. [80] F. Al-Agl, R. Brown, R. Steiner, Multiple categories: the equivalence between a globular and cubical approach, Adv. Math. 170 (2002), 71–118. [81] F. Al-Agl, Aspects of Multiple Categories, Ph.D. thesis, University of Wales, Bangor, 1989. [82] R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology (2009). [83] C.T.C. Wall, Finiteness conditions for CW -complexes II, Proc. Roy. Soc. Ser. A, 295 (1966), 149–166. [84] H.J. Baues, Algebraic Homotopy, volume 15 of Cambridge Studies in Advanced Mathematics, Cambridge Univ. Press, Cambridge, 1989. [85] J. Faria Martins, A new proof of a theorem of H.J. Baues, Preprint IST, Lisbon, 2007 p. 16. [86] E.J. Moore, Graphs of Groups: Word Computations and Free Crossed Resolutions, Ph.D. thesis, University of Wales, Bangor, 2001. [87] R. Brown, E. Moore, T. Porter, C. Wensley, Crossed complexes, and free crossed resolutions for amalgamated sums and HNN-extensions of groups, Georgian Math. J. 9 (2002), 623–644. [88] A. Heyworth, C.D. Wensley, IdRel - logged rewriting and identities among relators, GAP4, http://www.gap-system.org/Packages/idrel.html, version 2.05, 2008. [89] R. Brown, M. Golasi´nski, T. Porter, A. Tonks, Spaces of maps into classifying spaces for equivariant crossed complexes, Indag. Math. (N.S.) 8 (1997), 157–172. [90] R. Brown, M. Golasi´nski, T. Porter, A. Tonks, Spaces of maps into classifying spaces for equivariant crossed complexes. II. The general topological group case, K-Theory 23 (2001), 129–155.

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Hopf Algebraic Approach to Picard–Vessiot Theory Katsutoshi Amano, Akira Masuoka and Mitsuhiro Takeuchi Institute of Mathematics, University of Tsukuba, Ibaraki 305-8571, Japan E-mails: [email protected]; [email protected]; and [email protected]

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I: PV theory in the differential context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Formalism of Picard–Vessiot extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Galois correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Splitting ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Unique existence of minimal splitting ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Liouville extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II: PV theory in the C-ferential context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. C-ferential algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Formalism of PV extensions and Galois correspondence I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Galois correspondence II and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Uniqueness and existence of minimal splitting ﬁelds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part III: Uniﬁed PV theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. What’s the difference in the difference case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. AS D-module algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. PV extensions of AS D-module algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. Characterization and unique existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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128 130 130 135 139 143 144 149 149 152 153 158 159 159 161 164 168 170

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Introduction Picard–Vessiot (PV) theory is a Galois theory of linear differential equations; see [1–4]. Let us quickly review its basics along the traditional approach. A ﬁeld K together with a single derivation is called a differential ﬁeld. Suppose that the characteristic ch K of K is zero, and the subﬁeld k, say, of constants in K is algebraically closed. Given a linear homogeneous differential equation ∂y = y(n) + a1 y(n−1) + · · · + an−1 y = 0

(ai ∈ K )

with coefﬁcients in K, there is a unique (up to isomorphism) minimal splitting extension L/K of differential ﬁelds. By deﬁnition, L has the same constant ﬁeld k as K, includes an n-dimensional k-subspace of solutions of ∂y = 0, and is generated by those solutions over K. Such an extension L/K is called a PV extension, just as the minimal splitting ﬁeld of a (separable) algebraic equation is called as a Galois extension. The group Aut dif (L/K) of differential automorphisms of L/K forms a linear algebraic group as a closed subgroup of GLn (k), the group of linear automorphisms of the solution space. Then, M → Aut dif (L/M) gives a 1–1 correspondence (Galois correspondence) between the intermediate differential ﬁelds in L/K and the closed subgroups in Aut dif (L/K). The extension L/K is said to be Liouville if there is a sequence K = L0 ⊂ L1 ⊂ · · · ⊂ Lr = L of differential ﬁelds in which each Li /Li−1 is a ﬁnite extension or a simple extension Li = Li−1 (xi ), where xi is an integral, or the exponential of an integral, of some element in Li−1 . This condition holds if and only if the connected component in Aut dif (L/K) containing 1 is solvable. Parallel results were obtained by replacing differential ﬁelds with partial differential ﬁelds [5], -ﬁelds [6], and ﬁelds with higher derivation [7]. On contrary, the Franke [8] in his attempt of difference PV theory encountered the following difﬁculty: it often happens that a difference equation with coefﬁcients in a difference ﬁeld, say K, cannot split in any difference ﬁeld extension L/K with the same constants. Here, a difference ﬁeld is a ﬁeld together with a single automorphism. More than 30 years later, van der Put and Singer [9] overcame the difﬁculty stated before by extending the framework from difference ﬁelds to simple total difference rings and thereby succeeded in establishing the desired difference PV theory. A Hopf algebraic approach to the PV theory was proposed by Takeuchi [10] in the context of C-ferential ﬁelds, where C is a cocommutative coalgebra with speciﬁed grouplike 1C . With C appropriately chosen, differential ﬁelds, partial differential ﬁelds, -ﬁelds, and ﬁelds with higher derivation all turn into examples of C-ferential ﬁelds. According to this approach, to each PV extension L/K (see below for deﬁnition) is associated a commutative Hopf algebra H over the constant ﬁeld k, which is called the PV Hopf algebra for L/K. Quite generally, commutative Hopf algebras are in 1–1 correspondence with afﬁne group schemes. The afﬁne group scheme G(L/K) := Sp H corresponding to H is called the PV group scheme for L/K. This use of commutative Hopf algebras (or afﬁne group schemes) instead of linear algebraic groups makes it possible to avoid certain dispensable assumptions on k and L/K. For example, the Galois correspondence is formulated as a 1–1 correspondence between the intermediate C-ferential ﬁelds in L/K and the afﬁne closed subgroup schemes in

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G(L/K), for which k may be arbitrary, and L/K may not be ﬁnitely generated. Of course, this result reduces to the Galois correspondence cited in the ﬁrst paragraph in the special situation there. Amano and Masuoka [11] pursued Takeuchi’s approach, to include difference PV theory as well, working with AS D-module algebras. Here, D denotes a split cocommutative Hopf algebra over a ﬁxed ground ﬁeld R; it is thus of the form D1 #RG, where G denotes the group of grouplikes in D, and D1 is the irreducible component containing 1. A D-module algebra is said to be AS, if it is Artinian as a ring, and is simple as a D-module algebra; such a D-module algebra K is necessarily the product K1 × · · · × K1 of ﬁnitely many copies of a D(G1 )-module ﬁeld K1 , where G1 ⊂ G is the subgroup of the stabilizers of a primitive idempotent in K and D(G1 ) = D1 #RG1 . If D1 is trivial and G is the free group on one generator, then an AS D-module algebra is precisely a simple total ring as in [9]. Roughly speaking, the special situation where G is trivial corresponds to the C-ferential context, but Amano and Masuoka [11] imposes throughout the assumption that D1 is of Birkhoff– Witt type, which is assumed in [10] only to a small extent. The article [11] does not include [10] as a special case but clariﬁes how the argument in the C-ferential context can be used to deal with the context of AS D-module algebras. We remark that prior to Amano and Masuoka [11], André [12] gave a uniﬁed approach to differential and difference PV theories from the viewpoint of noncommutative differential geometry. Liouville extensions are not discussed in [10–12]. Amano [13] characterized Liouville PV extensions in the context of AS D-module algebras, introducing the notion of Liouville afﬁne group schemes. The present article brings the three papers [10, 11, 13] together and is divided into the three parts: Part I: PV theory in the differential context Part II: PV theory in the C-ferential context Part III: Uniﬁed PV theory To avoid too much technical argument, we discuss in Part I everything in the differential context. We then explain how the results are generalized in the C-ferential context (Part II) and in the context of AS D-module algebras (Part III). Part I is fairly selfcontained, with almost full proofs given. Only minimal knowledge of Hopf algebras and afﬁne group schemes is assumed there. The language of tensor categories will be used only in Parts II and III. In what follows in the Introduction, we will work in the differential context for simplicity. One more feature of our Hopf algebraic approach is ﬁrst to formalize the notion of a PV extension intrinsically (see below), and then to characterize it as a minimal splitting extension of a linear differential equation, just as Emil Artin formalized a Galois extension as a separable normal extension, subsequently characterizing it as a minimal splitting extension of a separable algebraic equation. To explain our formalism of PV extensions, let L/K be an extension of differential ﬁelds with the same constant ﬁeld k. We deﬁne L/K to be a PV extension (Deﬁnition 1.8) if there is a (necessarily unique) differential subring A ⊂ L including K such that (a) L equals the quotient ﬁeld Q(A) of A and (b) the constants H := (A ⊗K A)0 in A ⊗K A generate the left (or right) A-module A ⊗K A. It is then proved that the natural map

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μ : A⊗k H → A⊗K A, μ(a⊗h) = (a⊗1)·h is an isomorphism so that H is a k-form of the left A-module A⊗K A. The natural A-coring structure on A⊗K A induces on H a structure of a commutative k-Hopf algebra; this H is precisely the PV Hopf algebra cited before. Moreover, A/K turns into a right H-Galois extension [14, Deﬁnition → A ⊗k H required by the 8.1.1] for which the canonical isomorphism A ⊗K A − deﬁnition coincides with the inverse of μ. Thus, our approach is based on an idea of Hopf–Galois theory. According to our deﬁnition, PV extensions need not be ﬁnitely generated. We prove that L/K is ﬁnitely generated PV if and only if L is a minimal splitting ﬁeld of some linear differential equation (or a system of such equations) over K (Theorem 3.11). Giving a system of linear differential equations over K is equivalent to giving a differential K-module of ﬁnite K-dimension. Giving an appropriate deﬁnition of a minimal splitting ﬁeld of a possibly inﬁnite-dimensional differential K-module, we extend the last result, characterizing PV extensions in general (Proposition 3.17). There is another known approach to the PV theory proposed by Deligne [15] called the Tannaka approach. It is based on the result that a ﬁnite-dimensional differential K-module V generates the neutral Tannaka category, which corresponds to the PV group scheme G(L/K), where L is a minimal splitting ﬁeld for V . We reformulate this result in the C-ferential context (Theorem 8.11) and give an elementary proof based on a fundamental result of Hopf–Galois theory. In this way, our approach links the traditional approach and the Tannaka approach. Throughout this article, rings, ﬁelds, and algebras are supposed to be commutative unless otherwise stated. Part I: PV theory in the differential context 1. Formalism of Picard–Vessiot extensions Deﬁnition 1.1. A differential ring (or ﬁeld ) is a ring (or ﬁeld) A with a given derivation : A → A, which is by deﬁnition is an additive operation such that (ab) = a b + ab , and then necessarily 1 = 0. The subring (or subﬁeld) of constants in a differential ring (or ﬁeld) A is deﬁned and denoted by A0 = {a ∈ A | a = 0}. The adjective “differential” will mean -stable or -preserving. Thus, a differential subring is a -stable subring. A differential ring map is a -preserving ring map. Given such a map i : K → A, the pair (A, i) is called a differential K-algebra. Lemma 1.2 (Extension lemma). Let A be a differential ring and let S ⊂ A be a multiplicative subset. Then, the localization S −1 A by S has a unique differential A-algebra structure. The derivation is a a s − as = (a ∈ A, s ∈ S). s s2

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In particular, if A is an integral domain, its quotient ﬁeld, which will be denoted by Q(A), has a unique differential A-algebra structure. Proposition 1.3 (see [16, Lemma 10.2.2]). Let K be a differential ﬁeld, and let A be a differential K-algebra. Then, K and A0 are K0 -linearly disjoint, that is, the multiplication K ⊗K0 A0 → A, x ⊗ a → xa is injective. We will denote iterated derivations by x(1) = x , x(2) = x , . . . , x(i+1) = (x(i) ) . Proposition 1.4 (Wronskian criterion, [1, Theorem 3.7]). Finitely many elements x1 , x2 , . . . , xn in a differential ﬁeld K are K0 -linearly independent if and only if the determinant x1 x2 ··· xn x2 ··· xn x1 w(x1 , x2 , . . . , xn ) = , .. .. .. . . . (n−1) (n−1) (n−1) x x ··· x 1

2

n

called the Wronskian, is nonzero. Let K be a differential ﬁeld, and let A = 0 be a differential K-algebra. The Kalgebra A ⊗K A is regarded as differential with respect to (a ⊗ b) = a ⊗ b + a ⊗ b . Similarly, A ⊗K A ⊗K A and A ⊗K A ⊗K A ⊗K A are regarded as differential K-algebras. The A-bimodule C = CA/K := A ⊗K A

(1.5)

together with the A-bimodule maps : C = A ⊗K A → A ⊗K A ⊗K A = C ⊗A C, (a ⊗K b) = a ⊗K 1 ⊗K b, ε : C = A ⊗K A → A,

(1.6)

ε(a ⊗K b) = ab

forms an A-coring; this means that and ε satisfy the coalgebra axioms. Note that both and ε are maps of differential K-algebras, which therefore induce K0 -algebra maps 0 and ε0 on the constants. Let H = (A ⊗K A)0 denote the K0 -subalgebra of constants in A ⊗K A. Proposition 1.7. (1) We have three natural K0 -algebra maps: (i) K0 → A0 ,

(ii) H ⊗K0 H → (A ⊗K A ⊗K A)0 , and

(iii) H ⊗K0 H ⊗K0 H → (A ⊗K A ⊗K A ⊗K A)0 .

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(2) If both the maps (i) and (ii) are isomorphic, 0 and ε0 can be transfered to K0 -algebra maps, : H → H ⊗K0 H,

ε : H → K0 .

(3) If in addition, the map (iii) is injective, then (H, , ε) forms a commutative K0 -Hopf algebra. Its antipode S : H → H is induced by the ﬂip τ : A ⊗K A → A ⊗K A,

τ(a ⊗ b) = b ⊗ a.

In summary, (H, , ε, S ) is a kind of K0 -form of (CA/K , , ε, τ).1 Proof. To see (3), draw ﬁrst the commutative diagrams, which represent that CA/K is an A-coring, and take constants ( )0 . What is obtained are the commutative diagrams, which show that H is a K0 -coalgebra. To see then that S = τ 0 is a left, say, convolution-inverse of the identity map id H , take constants in the two coinciding maps

τ⊗id

id⊗ε

→ A ⊗K A ⊗K A −−−→ A ⊗K A ⊗K A −−→ A ⊗K A, A ⊗K A − ε

1⊗

→ A −→ A ⊗K A. A ⊗K A −

Deﬁnition 1.8. An inclusion L ⊃ K of differential ﬁelds will be called an extension and will be denoted by L/K. Such an extension L/K is called a PV (Picard–Vessiot) extension if the following are satisﬁed: (i) L0 = K0 . (ii) There exists an intermediate differential ring L ⊃ A ⊃ K such that (a) L = Q(A), the quotient ﬁeld of A, (b) H := (A ⊗K A)0 generates the left A-module A ⊗K A, that is, (A ⊗K K)·H = A ⊗K A; this is equivalent to the statement that H generates the right A-module A ⊗K A, as is seen by applying the ﬂip τ. Such an A as above is called a principal differential ring for L/K; it will be seen to be unique in 1.11. When L/K is PV, we will write k := L0 = K0 . Lemma 1.9. Let A be a principal differential ring for a PV extension L/K. Let H := (A ⊗K A)0 . (1) The natural maps (i)–(iii) of 1.7 are all isomorphisms so that H forms a commutative k-Hopf algebra. (2) We have a unique K-linear map a0 ⊗ a1 θ : A → A ⊗k H, θ(a) =

1 This means that tensoring up (H, , ε, S) with K over K gives (up to isomorphism) (C A/K , , ε, τ) 0

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such that for every a ∈ A, 1 ⊗K a = (a0 ⊗K 1)·a1 in A ⊗K A. (3) We have (i) The θ above is a k-algebra map, which makes A a right H-comodule; (A, θ) is thus a right H-comodule algebra. (ii) The k-subalgebra Aco H = {a ∈ A | θ(a) = a ⊗ 1} of H-coinvariants in A coincides with K. (iii) The left A-linearization of θ Aθ

: A ⊗K A → A ⊗k H,

A θ(a ⊗K

b) = aθ(b)

is an isomorphism. In summary, A/K is a right H-Galois extension [14, Deﬁnition 8.1.1] in the sense of Kreimer and Takeuchi [17]. Proof. (1) We have K0 = A0 , since k = K0 ⊂ A0 ⊂ L0 = k. It then sufﬁces to prove that the maps (ii) and (iii) of 1.7 are isomorphic. By applying 1.3 to the differential L-algebra L ⊗K A together with (L ⊗K A)0 ⊃ H, we see that the natural map L ⊗k H → L ⊗K A is injective. Hence, μ : A ⊗k H → A ⊗K A,

μ(a ⊗k h) = (a ⊗K 1)·h,

(1.10)

being surjective by assumption (ii)(b) of 1.8, is isomorphic. As an iteration of μ, we have

→ A ⊗K A ⊗k H − → A ⊗K A ⊗K A, A ⊗k H ⊗k H −

→ A ⊗K A ⊗K A ⊗K A. By taking ( )0 in these and similarly A ⊗k H ⊗k H ⊗k H − isomorphisms of differential K-algebras, the desired result follows. (2) Let θ be the composite of 1⊗ : A → A ⊗K A, a → 1 ⊗ a with μ−1 ; this is the unique map as required earlier. To see that (A, θ) is a right H-comodule, transfer the commutative diagrams, which show that the map 1⊗ : A → A ⊗A CA/K above satisﬁes the axioms for right comodules. (3) Note that A θ is an isomorphism with inverse μ. Lemma 1.11. A principal differential ring for a PV extension L/K is unique. Proof. Suppose we have two such rings, say A1 and A2 . Since A1 A2 is principal as well, we may suppose A1 ⊂ A2 . Then, the corresponding Hi := (Ai ⊗K Ai )0 gives an extension H1 ⊂ H2 of commutative Hopf algebras, which is necessarily faithfully ﬂat [18, Theorm 3.1]. The isomorphisms L ⊗k Hi L ⊗K Ai , which arise from the μ in (1.10), commute with the inclusions. It follows that A1 ⊂ A2 is faithfully ﬂat. Since A2 ⊂ Q(A1 ), we must have A1 = A2 . In summary, given a PV extension L/K, we have a unique principal differential ring A, which naturally deﬁnes a commutative k-Hopf algebra H = (A ⊗K A)0 .

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Deﬁnition 1.12. This H is called the PV Hopf algebra of L/K. We say that (L/K, A, H ) is a PV extension, indicating A, H too. Remark 1.13. Suppose that L/K is an extension of differential ﬁelds with the same constant ﬁeld k. Let A be a differential intermediate ring in L/K, and let J ⊂ (A⊗K A)0 be a k-subspace. As is seen from the proof of 1.9, if (A ⊗K K)·J = A ⊗K A, then J = (A ⊗K A)0 . Therefore, if L = Q(A) in addition, (L/K, A, J ) is PV. To give examples of PV extensions, recall from (1.5) that given a ﬁeld extension L/K, we have an L-coring CL/K = L ⊗K L. Just as with ordinary coalgebras, an element c in CL/K is called primitive (resp. grouplike), if (c) = 1 ⊗L c + c ⊗L 1, ε(c) = 0 (resp., (c) = c ⊗L c, ε(c) = 1). Lemma 1.14. In CL/K , (1) every primitive is of the form 1 ⊗K x − x ⊗K 1, where x ∈ L, (2) every grouplike is of the form x−1 ⊗K x, where 0 = x ∈ L. This is veriﬁed directly. One may note that the lemma can be reformulated as the vanishing of the Amitsur cohomology groups H 1 (L/K, Ga ), H 1 (L/K, Gm ) [19, Chapter 17]. The lemma yields the following. Example 1.15. Let L/K be an extension of differential ﬁelds with the same constant ﬁeld k. 2 (1) An element x ∈ L is said to be primitive over K, if x ∈ K or, symbolically speaking, if x is the integral a of some element a in K. This condition is equivalent to

l := 1 ⊗K x − x ⊗K 1 ∈ (L ⊗K L)0 . With x and l as given earlier, construct the subalgebras A = K[x] in L and H = k[l] in (A ⊗K A)0 ; note that A ⊂ A. Since 1 ⊗K x = x ⊗K 1 + l in A ⊗K A, we see that (K(x)/K, A, H ) is PV. Moreover, l is a primitive in H, and the map θ : A → A ⊗k H is given by θ(x) = x ⊗ 1 + 1 ⊗ l. Any PV Hopf algebra that is generated by a primitive arises in this way. (2) A nonzero element x in L is said to be exponential over K, if x x−1 ∈ K or, symbolically speaking, if x is the exponential exp a of some integral a (a ∈ K). This condition is equivalent to g := x−1 ⊗K x ∈ (L ⊗K L)0 . 2 Warning! One should distinguish the word “primitive” in this context of differential calculus, which comes from “primitive functions,” from the (more familiar) usage of the same word for primitive elements in a Hopf algebra or a coring, especially because these two distinct (though related) concepts can appear at the same time. Recall this warning at Deﬁnition 5.3, Theorem 5.8 and just above Theorem 8.4.

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With x and g as given earlier, construct A = K[x, x−1 ] in L and H = k[g, g−1 ] in (A⊗K A)0 ; note that A ⊂ A. Since 1⊗K x±1 = (x±1 ⊗K 1)·g±1 in A⊗K A, we see that (K(x)/K, A, H ) is PV. Moreover, g is a grouplike in H, and the map θ is given by θ(x±1 ) = x±1 ⊗ g±1 . Any PV Hopf algebra that is generated by a grouplike and its inverse arises in this way. To generalize the last example, we will work in the matrix algebra Mn (L ⊗K L), which is a noncommutative differential ring with entrywise derivation. Example 1.16. Let L/K and k be as given earlier. Fix an integer n > 0. Suppose that an invertible matrix X = (xij ) in GLn (L) is GLn -primitive in the sense that X X−1 ∈ Mn (K), that is, X = CX for some C ∈ Mn (K). Then, (X−1 ) = −X−1 C. Hence the matrices in Mn (L ⊗K L) B := (X−1 ⊗K 1)(1 ⊗K X),

B−1 := (1 ⊗K X−1 )(X ⊗K 1),

1 which are mutual inverses, are constants. Let B = (bij ), and construct A = K[xij , |X| ]

1 ] in (A ⊗K A)0 ; note that A ⊂ A. It follows from 1 ⊗K X = in L, and H = k[bij , |B| (X ⊗K 1)·B that (K(xij )/K, A, H ) is PV . Moreover,

(B) = (B ⊗k 1)(1 ⊗k B), θ(X) = (X ⊗k 1)(1 ⊗k B),

ε(B) = I,

S(B) = B−1 ,

where the last identity, for example, should be read as θ(xij ) = r xir ⊗k brj . We will see in Section 3 that any ﬁnitely generated PV extension arises in this way. 2. Galois correspondence To establish a Galois correspondence for a PV extension, let L/K be an extension of differential ﬁelds. The L-coring CL/K = L ⊗K L with its natural derivation may be called a differential L-coring. If M is an intermediate differential ﬁeld in L/K, then L ⊗M L is a quotient differential L-coring of CL/K , or in other words the kernel, say aM , of the natural surjection L ⊗K L L ⊗M L is a differential coideal in CL/K . Lemma 2.1. M → aM gives a bijection from the set of all intermediate differential ﬁelds M in L/K onto the set of all differential coideals a in CL/K . The inverse assigns to each a Ma := {x ∈ L | 1 ⊗K x − x ⊗K 1 ∈ a}.

(2.2)

This follows immediately from the correspondence due to Sweedler [20, Theorem 2.1] in the nondifferential situation. Proposition 2.3. Let (L/K, A, H ) be a PV extension.

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(1) There is a 1–1 correspondence between the ideals a0 in H and the differential ideals a in L ⊗K L, given by a0 = a ∩ H,

a = a0 ·(L ⊗K L).

(2) Suppose a0 ↔ a under this correspondence. Then a0 is a Hopf ideal if and only if a is a differential coideal. Proof. (1) Let I (resp., Idif ) indicate the set of ordinary (resp., differential) ideals. → L ⊗K A of differential It follows from the isomorphism L ⊗A μ : L ⊗k H − → Idif (L ⊗K A). This L-algebras that a0 → a0 ·(L ⊗K A) gives a bijection I(H ) − means that a0 → a0 ·(A ⊗K A) gives an injection I(H ) → Idif (A ⊗K A) with image Idif (L ⊗K A), which is contained in Idif (A ⊗K A), since L ⊗K A is a localization of A ⊗K A. By symmetry, the image above coincides with Idif (A ⊗K L) and hence with the intersection Idif (L ⊗K A) ∩ Idif (A ⊗K L) = Idif (L ⊗K L). (2) Similarly, we have a bijection I(H ⊗k H ) Idif (L ⊗K L ⊗K L) under which b0 := H ⊗k a0 + a0 ⊗k H ↔ b := L ⊗K a + a ⊗K L when a0 ↔ a under I(H ) Idif (L ⊗K L). Moreover, (a0 ) ⊂ b0 , ε(a0 ) = 0 if and only if (a) ⊂ b, ε(a) = 0; this proves (2). Here, recall that every biideal in a commutative Hopf algebra is stable under the antipode, as is seen in our situation from the fact that every coideal in L ⊗K L is stable under τ. Deﬁnition 2.4. A differential ring A = 0 is said to be simple if it includes no nontrivial differential ideal. In this case, A0 is a ﬁeld since every constant generates a differential ideal. Proposition 2.5. The principal differential ring A for a PV extension L/K is simple. Proof. Let 0 = a ⊂ A be a differential ideal. Suppose a0 ↔ L ⊗K a under I(H ) Idif (L ⊗K A). Then, a0 ·(L ⊗K L) = L ⊗K aL = L ⊗K L. By 2.3 (1), a0 = H, whence a = A. Theorem 2.6 (Galois correspondence I). Let (L/K, A, H ) be a PV extension. (1) There is a 1–1 correspondence between the Hopf ideals I in H and the intermediate differential ﬁelds M in L/K, given by M = {x ∈ L | 1 ⊗K x − x ⊗K 1 ∈ I·(L ⊗K L)}

(2.7)

I = H ∩ Ker(L ⊗K L L ⊗M L). (2) If I ↔ M under the correspondence, (L/M, AM, H/I) is a PV extension. Proof. (1) This follows from 2.1 and 2.3. (2) We see that

→A ⊗K A → AM·(H/I) = AM ⊗M AM, by considering the image of A ⊗k H − μ L ⊗M L.

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In general, let H be a commutative Hopf algebra over a ﬁeld k. Given a Hopf subalgebra J ⊂ H, the ideal HJ + generated by J + = Ker(ε : J → k) is a normal Hopf ideal in H; recall that a Hopf ideal I in H is called normal if a1 S(a3 ) ⊗ a2 ∈ H ⊗k I for all a ∈ I. It is known that J → HJ + gives a bijection from the set of all Hopf subalgebras in H onto the set of all normal Hopf ideals in H; see [18, Theorem 4.3]. Suppose I ↔ M under the correspondence in 2.6. It has been proved that the Hopf ideal I is normal if and only if M/K is PV. In other words, we have the following. Theorem 2.8 (Galois correspondence II). Let (L/K, A, H ) be a PV extension. There is a 1–1 correspondence between the Hopf subalgebras J in H and the intermediate PV extensions M/K in L/K, in which to each M/K its PV Hopf algebra is assigned. Proof. Fix a normal Hopf ideal I in H. Given a right H-comodule V with structure map λ : V → V ⊗k H, deﬁne V1 := {v ∈ V | λ(v) − v ⊗ 1 ∈ H ⊗k I}. When V = H, H1 is precisely the Hopf subalgebra corresponding to I [18, Corollary 3.10]. In general, V1 is the largest H1 -comodule in V or V1 = λ−1 (V ⊗k H1 ).

→ A⊗k H Since θ(A1 ) ⊂ A1 ⊗k H1 , we remark that the isomorphism A θ : A⊗K A − maps A1 ⊗K A1 into A1 ⊗k H1 . Regard A ⊗K A as a right H-comodule with respect to the structure arising from the right factor. Since the inclusion H → A ⊗K A is right H-colinear, H1 ⊂ A ⊗K A1 . By applying the antipode S and the ﬂip τ, we have H1 ⊂ A1 ⊗K A, whence H1 ⊂ A1 ⊗K A1 . This together with the last remark implies → A ⊗K A restricts to that the isomorphism μ : A ⊗k H −

→ A1 ⊗K A1 . A1 ⊗k H1 −

(2.9)

Hence, (L1 /K, A1 , H1 ) is PV, where L1 = Q(A1 ); note that A1 ⊂ A1 , L1 ⊂ L1 . To see the correspondence I ↔ L1 , note from (2.9) that in A1 ⊗K A1 , (A1 ⊗K K)·H1+ equals the ideal generated by 1 ⊗K a − a ⊗K 1 (a ∈ A1 ), whence in L ⊗K L, I·(L ⊗K L) = H1+ · (L ⊗K L) = Ker(L ⊗K L L ⊗L1 L). Conversely, let (M/K, B, J ) be an intermediate PV extension. Obviously, B ⊂ A, → B ⊗k J induces A ⊗K B − → A ⊗k J, which J ⊂ H. We see that B θ : B ⊗K B − → A ⊗k H. It follows that B = θ −1 (A ⊗k J). The argument extends to A θ : A ⊗K A − in the preceding paragraph shows that HJ + ↔ M. Let (L/K, A, H ) be a PV extension. We denote by G(L/K) = Sp H the afﬁne k-group scheme corresponding to H; let us call this the PV group scheme of L/K. By deﬁnition, this is the functor, which associates to each commutative kalgebra R the group Alg(H, R) of all k-algebra maps H → R; see [19, Section 1.4].

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Let Autdif(A/K) denote the functor, which associates to each R the group Aut dif(A ⊗k R/K ⊗k R) of differential K ⊗k R-algebra automorphisms of A ⊗k R. It has been proved that the k-linear representation G(L/K) → GL(A) arising from θ : A → → Autdif(A/K) of group-functors; see [10, A⊗k H gives an isomorphism φ : G(L/K) − Theorem (A.2)]. Consequently, the algebraic group G(L/K)(k) of the rational points in k is isomorphic to the group Aut dif(A/K) of differential K-algebra automorphisms of A, which is isomorphic to the group Aut dif(L/K) for L, by 1.11. This last group plays the role of Galois groups in standard PV theory (see 2.12 below). Example 2.10. Let the situation and the notation be as in 1.16, but we suppose L = K(xij ). We have a closed embedding of afﬁne k-group schemes, α : G(L/K) → GLn ,

g → g(B) = (g(bij )).

If f ∈ Aut dif(A ⊗k R/K ⊗k R), then β(f ) := X−1 ·f(X) ∈ GLn (R), since this equals the image A f(B) of the differential A-algebra map A f : A ⊗K A → A ⊗k R, A f(a ⊗K b) = a·f(b) applied entry-wise to B = (X−1 ⊗K 1)·(1 ⊗K X). Thus, we have a closed embedding β : Autdif (A/K) → GLn . Since β(φ(id H )) = A θ(B) = B = α(id H ), the following diagram of group functors commutes by the Yoneda lemma: G(L/K)

φ Autdif (A/K) @ @ αR @ R β @ GLn

The Galois correspondence theorems 2.6 and 2.8 are interpreted in terms of group schemes as follows. Theorem 2.11. Let L/K be a PV extension with PV group scheme G(L/K). (1) For every intermediate differential ﬁeld M in L/K, the extension L/M is PV. The map M → G(L/M) gives a bijection from the set of all intermediate differential ﬁelds in L/K onto the set of all closed subgroup schemes in G(L/K). (2) An intermediate differential ﬁeld M in L/K is PV over K if and only if G(L/M) is normal in G(L/K). In this case, we have G(M/K) G(L/K)/G(L/M). Remark 2.12. In the situation of 2.10, Autdif (L/K) ( G(L/K)(k)) is naturally a linear algebraic group over k. If k is an algebraically closed ﬁeld of characteristic zero, we obtain from 2.11 a 1–1 correspondence between the intermediate differential ﬁelds in L/K and the closed subgroups in Aut dif (L/K), since the latter are in 1–1 correspondence with the closed afﬁne group schemes in G(L/K). This is precisely the Galois correspondence in standard PV theory.

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Lemma 1.9 (3) is interpreted as follows. Proposition 2.13. Let (L/K, A, H ) be a PV extension. The k-scheme map Sp θ : Sp A ×k G(L/K) → Sp A corresponding to θ makes Sp A a right G(L/K)-torsor over Sp K. 3. Splitting ﬁelds Let K be a differential ﬁeld. Deﬁnition 3.1. A differential K-module is a K-module V together with an additive operation : V → V such that (av) = a v + av (a ∈ K, v ∈ V ). For such V , we write as before, V0 := {v ∈ V | v = 0}. This is a K0 -subspace of V . Proposition 1.3 can be directly generalized as follows. Proposition 3.2. For a differential K-module V , the natural map K ⊗K0 V0 → V , x ⊗ v → xv is injective. Lemma 3.3. Let V, W be differential K-modules. The K-module HomK (V, W ) of all K-linear maps V → W becomes a differential K-module with respect to f (v) := f(v) − f(v ) (f ∈ HomK (V, W ), v ∈ V ) in which HomK (V, W )0 coincides with the K0 -subspace HomK dif(V, W ) of differential K-module maps. By 3.2, we have K ⊗K0 HomK dif (V, W ) → HomK (V, W ). To give a variation, let L/K be an extension of differential ﬁelds, and suppose that W is a differential L-module. Then, HomK (V, W ) is a differential L-module so that we have L ⊗L0 HomK dif(V, W ) → HomK (V, W ). Important is the case when W = L. We then write LV := HomK dif(V, L). If the K-dimension dimK V = n < ∞, then the L0 -dimension dimL0 LV ≤ n, since we have L ⊗L0 LV → HomK (V, L).

(3.4)

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Example 3.5. A differential K-module V is said to be cyclic if it is generated by one element, say y; this means that V is K-spanned by y, y , y , . . . . If dimK V < ∞, such a pair (V, y) uniquely arises from a monic homogeneous linear differential equation with coefﬁcients in K, y(n) + a1 y(n−1) + · · · + an−1 y = 0 (ai ∈ K).

(3.6)

We recover V as V = Ky ⊕ Ky ⊕ · · · ⊕ Ky(n−1) with relation (3.6). By assigning the value f(y) to each f ∈ LV , we can identify LV with the L0 -space of the solutions of (3.6) in L, whose dimension is at most n. Example 3.7. The differential K-module V = Ky1 ⊕· · ·⊕Kyn of ﬁnite K-dimension, together with an ordered K-basis y1 , . . . , yn , uniquely arises from a homogeneous linear differential system with coefﬁcients in K, ⎞⎛ ⎞ ⎛ ⎞ ⎛ a11 a12 · · · a1n y1 y1 ⎟⎜ ⎟ ⎜ y ⎟ ⎜ a ⎜ 2 ⎟ ⎜ 21 a22 · · · a2n ⎟⎜ y2 ⎟ ⎟=⎜ ⎟⎜ ⎟ ⎜ (3.8) .. .. ⎟⎜ .. ⎟ (aij ∈ K ). ⎜ .. ⎟ ⎜ .. ⎝ . ⎠ ⎝ . . . ⎠⎝ . ⎠ yn

an1

an2

···

ann

yn

Here, (3.8) is regarded as giving the deﬁning relations for V . By assigning the vector to each f ∈ LV , we can identify LV with the L0 -space of soluwhose dimension is at most n. As is well known, the differential equation (3.6) can be treated as a differential system with coefﬁcient matrix ⎞ ⎛ 0 1 0 ··· 0 ⎜ .. ⎟ .. ⎟ ⎜ . 0 1 . ⎟ ⎜ ⎟ ⎜ ⎟. ⎜ .. .. ⎜ . . 0 ⎟ ⎟ ⎜ ⎟ ⎜ 0 1 ⎠ ⎝ −an−1 · · · · · · −a2 −a1

t (f(y ), . . . , f(y )) 1 n tions of (3.8) in Ln ,

Let L/K be an extension of differential ﬁelds. Given elements u1 , . . . , ur ∈ L, let Ku1 , . . . , ur denote the smallest intermediate differential ﬁeld in L/K containing u1 , . . . , ur . Given a differential K-module V , let KV denote the smallest intermediate differential ﬁeld in L/K including all f(V ), where f ∈ LV . We have KV V = LV . In 3.7, KV equals the smallest intermediate differential ﬁeld in L/K that contains all entries in the solutions of (3.8) in Ln . Lemma-Deﬁnition 3.9. Let L/K be as stated earlier. Let V be a differential K-module with dimK V = n < ∞. Then the following are equivalent: (i) L ⊗K V Ln as differential L-modules; (ii) The natural injection L ⊗L0 LV → HomK (V, L) is bijective; (iii) dimL0 LV = n.

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If these hold, L is called a splitting ﬁeld for V . If L = KV in addition, L is called a minimal splitting ﬁeld for V . Note that (i) implies KV ⊗K V KV n , whence KV is a splitting ﬁeld for V if L is. Proof. (ii) ⇔ (iii) is obvious. (i) implies LV Ln0 , (iii). Take the L-dual of the map in (ii). Then we have L ⊗K V → HomL0 (LV , L),

a ⊗ v → (f → af(v)).

(3.10)

Hence, (iii) implies (i).

Theorem 3.11 (Characterization). Let L/K be an extension of differential ﬁelds with the same constant ﬁeld k. Then the following are equivalent: (i) L/K is a PV extension, which is ﬁnitely generated in the sense L = Ku1 , . . . , un for certain ﬁnitely many elements u1 , . . . , un in L. (ii) L is a minimal splitting ﬁeld of some cyclic differential K-module of ﬁnite K-dimension. (iii) L is a minimal splitting ﬁeld of some differential K-module of ﬁnite Kdimension. (iv) There exists a GLn -primitive matrix X = (xij ) ∈ GLn (L) for some n (see 1.16), such that L = K(xij ). Proof. (ii) ⇒ (iii) is obvious. (iv) ⇒ (i) follows from 1.16. (i) ⇒ (ii). Suppose that (L/K, A, H ) is ﬁnitely generated PV. Suitable elements u1 , . . . , un as in (i) can be taken so that they form a k-basis of an H-subcomodule of A = (A, θ). Then, θ(uj ) = r ur ⊗k brj , where brj ∈ H. This means that in L ⊗K L, 1 ⊗K uj = r (ur ⊗K 1)·brj , whence (i)

1 ⊗K u j =

r

u(i) r ⊗K 1 ·brj

(i = 0, 1, . . . ) .

(3.12)

Let (i) W = uj ∈ GLn (L),

B = (bij ) ∈ Mn (H ),

(n) ∈ Ln . u(n) = u1 , . . . , u(n) n (3.13)

Note that W ∈ GLn (L) by 1.4. By (3.12), 1 ⊗K W = (W ⊗K 1) ·B,

1 ⊗K u(n) = (u(n) ⊗K 1)·B.

It follows that B ∈ GLn (H ). We have 1 ⊗K u(n) W −1 = (u(n) ⊗K 1)BB−1 (W ⊗K 1)−1 = u(n) W −1 ⊗K 1, whence u(n) W −1 ∈ Kn . Hence, there exist ai ∈ K such that u(n) = (a1 , . . . , an )W . Thus, u1 , . . . , un are solutions of y(n) = a1 y + · · · + an−1 y(n−1) ; this implies (ii).

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(iii) ⇒ (iv). Suppose that L is a minimal splitting ﬁeld of such a differential system as given by (3.8). Let x1 , . . . , xn (∈ Ln )

(3.14)

be k-linearly independent solutions of (3.8). The crucial point is that we see from (3.4) that they are L-linearly independent. One concludes that the matrix X = (x1 . . . xn ) satisﬁes Condition (iv). Corollary 3.15. (1) Let (L/K, A, H ) be a PV extension. The following are equivalent: (i) L/K is ﬁnitely generated as an extension of differential ﬁelds (see 3.11(i)); (ii) L/K is ﬁnitely generated as an extension of ﬁelds; (iii) A is ﬁnitely generated as a K-algebra; (iv) H is ﬁnitely generated as a k-algebra. (2) Any intermediate PV extension M/K in a ﬁnitely generated PV extension L/K is ﬁnitely generated. (3) If (L/K, A, H ) is a ﬁnitely generated PV extension, then the transcendence degree trans.degK L and the Krull dimensions Kdim A and Kdim H all coincide. Proof. (1) We see from the last proof that (i) ⇔ (ii) ⇔ (iii). The remaining equivalence (iii) ⇔ (iv) follows from L ⊗K A L ⊗k H. (2) This follows since a Hopf subalgebra of a commutative ﬁnitely generated Hopf algebra is ﬁnitely generated [18, Corollary 3.11]. (3) This also follows from L ⊗K A L ⊗k H. Let us extend the notion of (minimal) splitting ﬁelds so that the differential K-module in question can be of inﬁnite dimension. Lemma-Deﬁnition 3.16. Let L/K be an extension of differential ﬁelds. Let V be a differential K-module of possibly inﬁnite K-dimension. The following are equivalent: (i) There exists a differential L-linear injection L ⊗K V → λ∈ Lλ into the direct product of copies Lλ of L, (ii) The natural injection L ⊗L0 LV → HomK (V, L) has a dense image, or in other words, its L-dual given by (3.10) is injective. If these conditions (and L = KV ) hold, L is called a (minimal) splitting ﬁeld for V . Note from (i) that KV is a splitting ﬁeld for V if L is. Proof. Modify the proof of 3.9.

Proposition 3.17. Let L/K be an extension of differential ﬁelds with the same constant ﬁeld. Then, L/K is PV if and only if L is a minimal splitting ﬁeld for a differential

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K-module V , which is locally ﬁnite in the sense that it is a sum of differential Ksubmodules of ﬁnite K-dimension. Proof. “Only if”. Suppose that H is the PV Hopf algebra of L/K. One can write H = λ Hλ as a directed union of ﬁnitely generated Hopf subalgebras Hλ . We have L = λ Lλ , where Lλ /K is a ﬁnitely generated PV extension with Hopf algebra Hλ ; see 2.8, 3.15. By 3.11, Lλ is a minimal splitting ﬁeld of a differential K-module, say K-dimension, say n . Let V = this is locally ﬁnite. We have Vλ , of ﬁnite λ λ Vλ ; KV = λ KVλ = L and L ⊗K V λ Lnλ → λ Lnλ . “If”. Suppose that L is a minimal splitting ﬁeld for V , where V = ν Vν is a Then, Lν := directed union of differential K-submodules Vν of ﬁnite K-dimension. . KVν (⊂ L) is a minimal splitting ﬁeld for Vν such that L = ν L ν By 3.11, we have ﬁnitely generated PV extensions, (Lν /K, Aν , Hν ). Let A := ν Aν ; this is a directed union of intermediate differential rings in L/K, with L = Q(A). Bytaking ν in Aν ⊗k Hν Aν ⊗K Aν , we obtain A ⊗k H A ⊗K A, where H = ν Hν . It follows that (L, A, H ) is PV.

4. Unique existence of minimal splitting ﬁelds Let A be a ring. Let A[[t]] denote the ring of formal power series of one variable t with coefﬁcients in A. This is regarded as a differential ring with respect to the standard derivation ∞ ∞ n an t = nan t n−1 (an ∈ A). n=0

n=1

Given an ideal I of A, I[[t]] =

∞

n=0 an t

n

| an ∈ I is a differential ideal of A[[t]].

Lemma 4.1 (cf. [1, Lemma 1.8, Theorem 2.1]). Suppose that A is a differential ring including Q as a subﬁeld. a(n) n (1) ρ : A → A[[t]], ρ(a) = ∞ n=0 n! t is an injection of differential rings. (2) An ideal I of A is a differential ideal if and only if√ I = ρ−1 (I[[t]]). (3) If I is a differential ideal of A, then the radical I of I is differential too. √ Moreover, I equals the intersection λ Qλ of suitable differential ideals Qλ that are prime. (4) If A is simple as a differential ring (see 2.4), it is an integral domain. Proof. √(1), (2) Easy. (2), since I = ρ−1 (I[[t]]) √ (3) The ﬁrst assertion follows from −1 −1 For the second, let Qλ = ρ (Pλ [[t]]), where Pλ are implies I = ρ ( I[[t]]). √ prime ideals such that I = λ Pλ . (4) This follows from (3). Lemma 4.2 (Levelt [21, Appendix]). Let K be a differential ﬁeld of characteristic ch K = 0, and let A be a differential K-algebra. Suppose that A is ﬁnitely generated

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as a K-algebra and is simple as a differential ring, which is then necessarily an integral domain; see 4.1 (4). Let L = Q(A); see 1.2. Then the ﬁeld extension L0 /K0 is algebraic. Proof. First, we claim L0 = A0 . In fact, if x ∈ L0 , then {a ∈ A | ax ∈ A} is a nonzero differential ideal in A and hence equals A, whence x ∈ A0 . Next, we see that every y ∈ A0 is algebraic over K, since for any maximal ideal P of A, the constant ﬁeld A0 is included in A/P, which, being ﬁnitely generated as a K-algebra, is algebraic over K. Let ϕ(t) = t n + c1 t n−1 + · · · + cn be the minimal polynomial of y (∈ A0 ) over K. Since c1 t n−1 + c2 t n−2 + · · · + cn has y as a root, the minimality of ϕ(t) implies that all ci ∈ K0 so that y is algebraic over K0 . Theorem 4.3 (Unique existence). Suppose that K is a differential ﬁeld of characteristic 0 whose constant ﬁeld K0 is algebraically closed. Let V be a differential K-module of ﬁnite K-dimension. Then, there exists a minimal splitting ﬁeld L for V with L0 = K0 so that L/K is necessarily a ﬁnitely generated PV extension. Such an L is unique up to isomorphism of differential K-algebras. Proof. Existence. Suppose that (3.8) is the differential system corresponding to V with respect to an ordered K-basis y1 , . . . , yn . Let K[Xij ] be the polynomial algebra 1 ] be its localization at the determinant in Xij (1 ≤ i, j ≤ n), and let O := K[Xij , det det(Xij ). Make these into differential K-algebras by deﬁning (Xij ) = (aij )(Xij ). Choose a maximal differential ideal P in O, and let L = Q(O/P). By 4.2, L0 = K0 , since K0 is algebraically closed. Let xij denote the natural image of Xij in L. Then, L = K(xij ), and the matrix (xij ) is invertible. It follows that t (x1j , . . . , xnj ) (1 ≤ j ≤ n) are n K0 -linearly independent solutions of (3.8). Hence, L is what is required. Uniqueness. Suppose we have two ﬁelds, Li (i = 1, 2), such as required. Let Ai be the principal differential ring for Li /K, and deﬁne A = A1 ⊗K A2 , and consider it as a differential K-algebra. Choose a maximal differential ideal I in A, and let L = Q(A/I). Again by 4.2, L0 = K0 . Since Ai is simple by 2.5, the natural maps Ai → A/I of differential K-algebras are both injective so that they extend → LV by to injections Li → L of differential ﬁelds over K. Since we have LVi − counting L0 -dimension, the images of the Li are both KV in L so that L1 L2 , as desired. 5. Liouville extensions ¯ A separable k-algebra is a ﬁniteLet k be a ﬁeld with an algebraic closure k. dimensional k-algebra R such that the base extension R ⊗k k¯ is reduced; this is equivalent to the statement that R is isomorphic to a product L1 × · · · × Lr of ﬁnitely many separable ﬁeld extensions Li /k of ﬁnite degree. Every ﬁnitely generated k-algebra A contains the largest separable subalgebra; it will be denoted by π0 A, called the separable part of A. We have π0 (A⊗k B) = π0 A⊗k π0 B, π0 (A⊗k k ) = π0 A⊗k k , where A and B are ﬁnitely generated k-algebras, and k /k is a ﬁeld extension.

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If H is a ﬁnitely generated commutative Hopf algebra, then π0 H ⊂ H is a ﬁnitedimensional Hopf subalgebra. Let H 0 := H/(π0 H )+ H denote the corresponding quotient Hopf algebra. Note that (π0 H )+ H is the smallest ideal I of H such that ε(I) = 0, and H/I is coconnected in the sense that the topological space Spec H/I of all prime ideals is connected; this means that H/I contains no nontrivial idempotents. If k is algebraically closed, π0 H is isomorphic to the dual (k)∗ of the group algebra k of some ﬁnite group . Proposition 5.1. Let (L/K, A, H ) be a ﬁnitely generated PV extension with k = K0 (= L0 ). (1) The K-separable part π0 A of A is an intermediate differential ﬁeld in L/K. It coincides with the separable algebraic closure Ks of K in L. In particular, Ks /K is a ﬁnite extension. (2) (π0 A/K, π0 A, π0 H ) is a PV extension. Suppose that k is algebraically closed so that π0 H = (k)∗ with a ﬁnite group. Then, π0 A/K is a ﬁnite Galois extension with Galois group . (3) L/K is ﬁnite separable if and only if H is separable as a k-algebra. (4) L/K is separably closed (i.e. Ks = K) if and only if H is coconnected. Proof. (1), (2) Taking the K-separable part in A⊗K A A⊗k H = A⊗K (K⊗k H ), we have π0 A ⊗K π0 A π0 A ⊗ π0 H,

(5.2)

whence A ⊗K π0 A A ⊗k π0 H. This implies π0 A = θ −1 (A ⊗k π0 H ). Hence, π0 A is an intermediate differential ﬁeld in L/K. Part 2 now follows by (5.2). Set M = π0 A. It remains to prove that L/M is separably closed. Since (L/M, A, H 0 ) is PV by 2.6 (2) and 2.8, we have L ⊗M A L ⊗k H 0 . Since the base extension L ⊗k H 0 is still coconnected, its spectrum contains the nilradical as a generic point. Therefore, its localization L⊗M L, whose spectrum necessarily contains such a point, is coconnected; this implies what should be proved. (3), (4) These follow from (1) and (2). Deﬁnition 5.3. An extension L/K of differential ﬁelds is said to be Liouville if the following are satisﬁed. (i) L0 = K0 . (ii) There is a sequence of intermediate differential ﬁelds in L/K K = L0 ⊂ L1 ⊂ · · · ⊂ Lr = L in which each Li /Li−1 (0 < i ≤ r) is either (a) ﬁnite separable, (b) of the form Li = Li−1 (x) with x primitive over Li−1 (1.15 (1)), or (c) of the form Li = Li−1 (x) with x exponential over Li−1 (1.15 (2)). Of course it follows that L/K is ﬁnitely generated as a ﬁeld extension.

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To characterize Liouville PV extensions, the language of group schemes seems more suitable than that of Hopf algebras. We will use some terminology and basic results on group schemes from [19]. Let G = Sp H be an afﬁne group scheme over a ﬁeld k, which corresponds to a commutative Hopf algebra H. Suppose that H is ﬁnitely generated, or in other words, G is algebraic. The dimension dim G is by deﬁnition the Krull dimension Kdim H. We write π0 G = Sp(π0 H ),

G0 = Sp H 0 .

The latter is called the connected component of the unit element in G. We have a short exact sequence G0 G π0 G of afﬁne group schemes. G is said to be ﬁnite etale (resp., connected) if G = π0 G (resp., G0 = G). If k is algebraically closed, G0 G splits as a left or right G0 -equivariant morphism of schemes. Example 5.4 (Two basic afﬁne group schemes). (1) The additive group Ga = Sp k[X]; X is a primitive in k[X]. The group Ga (R) of the rational points in a kalgebra R is the additive group R. A closed subgroup G of Ga corresponds to a Hopf algebra H = k[l] that is generated by a primitive l, with a closed embedding given by X → l, k[X] k[l]. If G = Ga , it is connected. Suppose that G Ga . If r ch k = 0, then G = {1}. If ch k = p > 0, then H = k[X]/(f(Xp )), where r ≥ 0 2 s and f(X) = X + c1 Xp + c2 Xp + · · · + cs Xp (ci ∈ k). In this case, r

H 0 = k[X]/(Xp ),

r

r

π0 H = k[Xp ]/(f(Xp )).

(2) The multiplicative group Gm = Sp k[X, X−1 ]; X is a grouplike in k[X, X−1 ]. The group Gm (R) is the group of units in R. A closed subgroup of Gm corresponds to a Hopf algebra H = k[g, g−1 ] that is generated by a grouplike g and its inverse, with a closed embedding given by X±1 → g±1 , k[X, X−1 ] k[g, g−1 ]. If G = Gm , it is connected. Suppose that G Gm . Then, G is ﬁnite and g has a ﬁnite order, say n. If ch k = 0, G is ﬁnite etale. Suppose ch k = p > 0 and n = pe n with p n . In this case, e

H 0 = k[X]/(Xp − 1),

e

π0 H = k[Xp ]/(Xn − 1).

Deﬁnition 5.5. An algebraic afﬁne group scheme G is said to be Liouville, if there is a normal chain of closed subgroups G = G0 G1 · · · Gr = {1} in which each factor Gi /Gi+1 (0 ≤ i < r) is either (a) ﬁnite etale, (b) a closed subgroup scheme of Ga , or (c) a closed subgroup scheme of Gm . Such a chain as above will be called a Liouville normal chain (LNC) of length r.

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Lemma 5.6. Let G be an algebraic afﬁne group scheme over a ﬁeld k. (1) If G is Liouville, every closed subgroup of G is Liouville. (2) Given a normal closed subgroup N G, G is Liouville if and only if N and G/N are both Liouville. (3) G is Liouville if and only if G0 is Liouville. Proof. (1), (2) Standard; see the proof of [13, Lemma 1.2]. (3) This follows from (2). Proposition 5.7. Let G be a connected algebraic afﬁne group scheme over a ﬁeld k. (1) Suppose that G is Liouville. Then, it has an LNC in which each factor is a connected closed subgroup of Ga or Gm . If ch k = 0, this means that an LNC can be chosen so that each factor is isomorphic to Ga or Gm . (2) If G is Liouville, it is solvable (i.e. has a normal chain with Abelian factors). The converse holds if k is algebraically closed. Proof. (1) Let G = G0 · · · Gr = {1} be an LNC of least length. We will proceed by induction on r. Since G is connected, the derived closed subgroup DG is connected as well [19, Theorem 10.1]. The ﬁrst factor G/G1 cannot be ﬁnite etale so that DG ⊂ G1 . Hence, the least length of LNC’s for DG is smaller than r; the induction hypothesis can apply to DG. By replacing G with G/DG, we may suppose that G is Abelian. We then have a short exact sequence Gs G Gu , where Gu is unipotent and Gs is of multiplicative type; this sequence splits if k is perfect [19, Theorem 9.5]. So it remains to prove (1) when G = Gu or Gs . The case G = Gu . If ch k = 0, G Ga × · · · × Ga . In general, every (connected) unipotent algebraic afﬁne group scheme has a normal chain in which every factor is a (connected) closed subgroup of Ga . Hence we are done. The case G = Gs . Since G is connected, it is a torus if ch k = 0. If ch k = p > 0, we have a short exact sequence T G F, where T is a torus, and F (= Sp J) is a ﬁnite group scheme of multiplicative type whose order, i.e. the dimension dim J, is a power of p. The desired result for F is proved by induction on its order. In any characteristic, we may suppose that G is a torus. The ﬁrst factor G/G1 then must be a closed subgroup of Gm . The result applied to the largest anisotropic subtorus Ta in G implies that Ta must be trivial. By [19, Theorem 7.4], this means that G splits or G Gm × · · · × Gm . The desired result is then obvious. (2) The ﬁrst assertion follows by (1). For the second, we can replace G with Di G/Di+1 G, and may suppose that G is Abelian, in which case we are done using the argument stated earlier since Gs is diagonalizable if k is algebraically closed. Theorem 5.8. Let L/K be a ﬁnitely generated PV extension with PV group scheme G = G(L/K). Let Ks denote the separable algebraic closure of K in L; see 5.1 (1).

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(1) The following are equivalent: (i) L/K is Liouville. (ii) There is a sequence of intermediate differential ﬁelds in L/Ks Ks = L0 ⊂ L1 ⊂ · · · ⊂ Lr = L

(5.9)

such that for each 0 < i ≤ r, Li−1 is separably closed in Li , and Li = Li−1 (xi ), where xi is either primitive or exponential over Li−1 . ˜ (iii) L/K is embedded into some Liouville extension L/K. (iv) G is Liouville. (v) G0 is Liouville. If the constant ﬁeld k is algebraically closed, these are further equivalent to (vi) G0 is solvable. (2) Suppose ch k = 0, and that L/K is Liouville. Then, such a sequence as in (ii) earlier can be chosen so that for each 0 < i ≤ r, xi is transcendental over Li−1 or equivalently G(Li /Li−1 ) Ga or Gm . In this case, r = trans.degK L = dim G = the least length of LNC’s for G0 . Proof. (1) Recall from 1.15 the characterization of the two types of basic PV extensions. Then, Theorem 2.11, combined with 5.1 and 5.7, proves the equivalence among the conditions except (iii). Obviously, (i) ⇒ (iii). Suppose (iii), and that ˜0 ⊂ L ˜1 ⊂ ··· ⊂ L ˜r = L ˜ is a sequence satisfying the conditions in 5.3. We K=L will prove (iv) and (v) by induction on r. If r = 0, this is obvious. Suppose r > 0, and ˜ 1 /L ˜ 1 is a ﬁnitely gener˜ 1 be the composite ﬁeld. By the next proposition, LL let LL ˜ 1 /L ˜ 1 ) G(L/L ∩ L ˜ 1 ). With the induction hypothesis ated PV extension with G(LL ˜ 1 , G(L/L ∩ L ˜ 1 ) is Liouville. If L ˜ 1 /K is ﬁnite separable, L/L ∩ L ˜1 ˜ 1 /L applied to LL ˜ 1 ), and (v) follows. Otherwise, L ∩ L ˜ 1 /K is a too is, so that G(L/K)0 ⊂ G(L/L ∩ L ˜ 1 /K), is a closed subgroup PV extension whose PV group scheme, a quotient of G(L of Ga or Gm ; (iv) follows. (2) This follows from Part 1 just proven and 5.7 (1). ˜ ˜ 0 = K0 . Proposition 5.10. Let L/K be an extension of differential ﬁelds such that L ˜ Let L, M be an intermediate differential ﬁelds in L/K. Suppose that L/K is a ﬁnitely generated PV extension. Then, LM/M is a ﬁnitely generated PV extension whose PV group scheme G(LM/M) is naturally isomorphic to G(L/L ∩ M). Proof. Set F := L ∩ M. Let X = (xij ) ∈ GLn (L) be a GLn -primitive matrix such that L = K(xij ); see 3.11 (iv). Set B = (bij ) := (X−1 ⊗F 1)(1 ⊗F X) in Mn (L ⊗F L), and B˜ = (b˜ ij ) := (X−1 ⊗M 1)(1 ⊗M X) in Mn (LM ⊗M LM). Construct the subalgebras 1 1 ˜ := k b˜ ij , H := k bij , ⊂ LM ⊗M LM ⊂ L ⊗F L, H ˜ |B| |B| over the constant ﬁeld k = K0 . By 3.11, L/F (resp., LM/M) is a ﬁnitely generated PV ˜ It remains to prove that the canonical extension with PV Hopf algebra H (resp., H). ˜ ˜ epimorphism ϕ : H → H, ϕ(bij ) = bij is injective. Consider the correspondence of

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2.1 applied to L/F . It then sufﬁces to prove that the intermediate differential ﬁeld Ma which corresponds to the differential coideal a := (Ker ϕ) · (L ⊗F L) coincides with F . Since a ⊂ Ker (L ⊗F L → LM ⊗M LM), it follows that if x Ma , then 1 ⊗M x = x ⊗M 1 in LM ⊗M LM, whence x L ∩ M = F ; this proves Ma = F , as desired.

Part II: PV theory in the C-ferential context Throughout this part, we work over a ﬁxed ground ﬁeld R. All vector spaces, (co)algebras, and linear maps are over R. Let ⊗, and Hom stand for the tensor product ⊗R over R, and the vector space HomR of linear maps, respectively. Let C denote a cocommutative coalgebra (over R) with a speciﬁed grouplike 1C . 6. C-ferential algebras We let : C → C ⊗ C,

(c) =

c1 ⊗ c2 ;

ε:C→R

denote the structure maps of C. So, (1C ) = 1C ⊗ 1C , ε(1C ) = 1. Deﬁnition 6.1. A C-ferential module is a vector space V equipped with a unital C-action ψ : C ⊗ V → V by which we mean that ψ is a linear map such that ψ(1C ⊗ v) = v (v ∈ V ). We will simply write cv for ψ(c ⊗ v). Let C M denote the category of C-ferential modules and C-ferential (i.e. C-action preserving) maps. The category C M forms an R-linear Abelian tensor category (C M, ⊗, R) with respect to the tensor product V ⊗ W (V, W ∈ C M) endowed with the C-action c(v ⊗ w) = c1 v ⊗ c2 w (c ∈ C, v ⊗ w ∈ V ⊗ W ) and the unit object R with the trivial action c1 = ε(c)1. This is symmetric with respect to the obvious symmetry v ⊗ w → w ⊗v, V ⊗ W →W ⊗ V , since C is c1 ⊗ c2 (c ∈ C)]. supposed to be cocommutative [or in notation, c2 ⊗ c1 = We attach the adjective ‘C-ferential’ to the algebra- and the module-objects in C M, as follows. Deﬁnition 6.2. A commutative algebra in the symmetric tensor category C M is called a C-ferential algebra; we thus keep the assumption that algebras are commutative. For such an algebra A, an A-module in C M is called a C-ferential A-module. Let A,C M denote the category of C-ferential A-modules and C-ferential A-linear maps. Explicitly, a C-ferential algebra is an ordinary commutative algebra A equipped with a unital C-action such that c(ab) = (c1 a)(c2 b), c1 = ε(c)1 (c ∈ C, a, b ∈ A).

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It is called a C-ferential ﬁeld if it is a ﬁeld. A C-ferential A-module is an ordinary A-module M together with a unital C-action such that c(am) = (c1 a)(c2 m) (c ∈ C, a ∈ A, m ∈ M). Remark 6.3. Let C+ = Kerε. The tensor algebra T(C+ ) on C+ has a unique (cocommutative) bialgebra structure such that the natural embedding C → T(C+ ), which sends 1C to the unit 1 of T(C+ ), and is the identity on C+ , is a coalgebra map. In the obvious way, a C-ferential module is identiﬁed with a left (or right) T(C+ )module. Moreover, the symmetric tensor category C M is identiﬁed with that of left T(C+ )-modules T(C+ ) M. A C-ferential algebra A and a C-ferential A-module are identiﬁed with a T(C+ )-module algebra and a left module over the smash-product algebra A#T(C+ ), respectively. Thus, A,C M = A#T(C+ ) M; see [16, Chapter VII]. Example 6.4. Suppose C = R1C ⊕ Rd, where d( = 0) is a primitive. Then, C M is precisely the tensor category of differential R-modules, where R is regarded as a differential algebra with zero derivation. Hence, C-ferential objects (e.g. C-ferential algebras) equal differential objects over R (e.g. differential R-algebras). Note that any differential ﬁeld K can be regarded as a C-ferential ﬁeld, by choosing as R the prime ﬁeld of K. One will see that the results in this part specialize to those in the previous part, with C chosen just as above, and also specialize to the situation of [6, p. 3], [7], with C chosen as in the following two examples. In Sections 8 and 9, we will impose some additional assumptions on C, which are, however, fulﬁlled for these three examples at least in characteristic zero. Example 6.5. (1) Suppose that C = R1C ⊕ δ∈ Rδ, where δ ( = φ) is a family of primitives3 . Then, a C-ferential algebra is precisely an algebra A given a family of linear derivations on A; this is a -algebra in the sense of Buium [6]. · · , where d0 = 1C , d1 , d2 , . . . form a (2) Suppose that C = R1C ⊕ Rd1 ⊕ Rd2 ⊕ · divided power sequence in the sense (dn ) = ni=0 di ⊗ dn−i (n = 1, 2, . . . ). Then, a C-ferential algebra is an algebra A provided with a higher derivation, that is, an inﬁnite sequence of linear endomorphisms ∂0 = id, ∂1 , ∂2 , . . . such that ∂n (ab) = n i=0 (∂i a)(∂n−i b) (a, b ∈ A). Given a C-ferential module V , we deﬁne the subspace of constants by V0 = {v ∈ V | cv = ε(c)v

(c ∈ C)}.

If A is a C-ferential algebra, A0 ⊂ A is a subalgebra. More generally, suppose that A is a C-ferential K-algebra, that is, a C-ferential algebra equipped with a C-ferential 3 Following a custum we have used the symbol to denote families of primitives and of derivations; the same symbol always denotes the comultiplication of a coalgebra, elsewhere in the text.

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algebra map K → A. Then A0 is a K0 -subalgebra of A. If M ∈ A,C M, then M0 ⊂ M is an A0 -submodule. Deﬁnition 6.6. A C-ferential algebra A ( = 0) is said to be simple if it includes no nontrivial C-ferential ideal. Proposition 6.7 (see [16, Lemma 10.2.2]). A C-ferential algebra A is simple if and only if (a) A0 is a ﬁeld and (b) for every M ∈ injective.

A,C M,

the natural map A ⊗A0 M0 → M, a ⊗ m → am is

In general, if D is a coalgebra and A an algebra, Hom(D, A) becomes an algebra with respect to the convolution product (f ∗ g)(d) = f(d1 )g(d2 ), where f, g ∈ Hom(D, A), d ∈ D, (d) = d1 ⊗ d2 ; see [16, Section 4.0]. Lemma 6.8. Suppose that A is a C-ferential algebra. Then Hom(T(C+ ), A) is a C-ferential A-algebra with respect to the C-action (cf )(h) = f(hc)

(c ∈ C, h ∈ T(C+ ), f ∈ Hom(T(C+ ), A))

and the (splitting) algebra map ρ˜ : A → Hom(T(C+ ), A),

a → (h → ha).

(6.9)

We have Hom(T(C+ ), A)0 = Hom(T(C+ )/Kerε, A) = A. Corollary 6.10. If K is a C-ferential ﬁeld, then the map K ⊗K0 K → Hom(T(C+ ), K),

a ⊗ b → (h → a(hb))

is injective. Proof. This follows from property (b) in 6.7 which holds for the C-ferential K-module Hom(T(C+ ), K). The following will play the role of the Wronskian criterion 1.4. Proposition 6.11. In a C-ferential ﬁeld K, ﬁnitely many, say n, elements a1 , . . . , an are K0 -linearly independent if and only if there exist n elements h1 , . . . , hn in T(C+ ) such that the matrix (hi aj ) in Mn (K) is invertible. Proof. “If ”. If j cj aj = 0 with ci ∈ K0 , j cj (haj ) = 0 (h ∈ T(C+ )), whence (c1 , . . . , cn )t (hi aj ) = (0, . . . , 0); this implies c1 = · · · = cn = 0.

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“Only if.” Set W =

n

j=1 K0 aj .

By 6.10, we have a K-linear injection

K ⊗K0 W → Hom(T(C+ ), K) HomK (K ⊗ T(C+ ), K), whose K-predual K ⊗ T(C+ ) → HomK0 (W, K) = Kn ,

a ⊗ h → (a(haj ))j

must be surjective. Hence, there exist h1 , . . . , hn ∈ T(C+ ) such that their restrictions h|W , . . . , hn |W form a K-basis of HomK0 (W, K), or in other words, (hi aj ) is invertible. 7. Formalism of PV extensions and Galois correspondence I Both the “PV extension formalism” and “Galois correspondence I”, [see Theorem 2.6] can be directly generalized to the C-ferential context, without any additional assumptions on C. Deﬁnition 7.1. An inclusion L ⊃ K, or an extension L/K, of C-ferential ﬁelds is called a PV extension, if (i) L0 = K0 , which will be denoted by k, and (ii) there exists an intermediate C-ferential algebra L ⊃ A ⊃ K such that (a) L = Q(A), the quotient ﬁeld of A, (b) A ⊗K A is generated by (A ⊗K A)0 as a left (or equivalently right) A-module. Such an A as given earlier is called a principal C-ferential algebra for L/K. With L/K, A as given earlier, A ⊗K A is naturally a C-ferential K-algebra so that H := (A ⊗K A)0 is an algebra over k = K0 . The A-bimodules in C M form a tensor category, (A (C M)A , ⊗A , A). The object A⊗K A in A (C M)A has the natural structure maps and ε given by (1.6), of a C-ferential A-coring (i.e. a coalgebra in the tensor category just mentioned). Just as in the proof of 1.9, the property (b) in 6.7 of the → A ⊗K A, C-ferential L-module L ⊗K A gives the isomorphism μ : A ⊗k H − μ(a ⊗k h) = (a ⊗K 1)·h, from which the next two results follow. Lemma 7.2. Let A be a principal C-ferential algebra in a PV extension L/K of Cferential ﬁelds. Let H := (A ⊗K A)0 . (1) The C-ferential K-algebra maps , ε, and the ﬂip τ induce on H structure maps : H → H ⊗k H,

ε : H → k,

S:H →H

which make it a commutative k-Hopf algebra.

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(2) A/K is a right H-Galois extension with respect to the H-comodule structure θ : A → A ⊗k H, θ(a) = a0 ⊗ a1 uniquely determined by 1 ⊗K a = (a0 ⊗K 1) · a1 in A ⊗K A. Lemma 7.3. For each PV extension of C-ferential ﬁelds, its principal C-ferential algebra is unique. Given a PV extension L/K, we say that (L/K, A, H ) is a PV extension, to indicate that A is the (unique) principal C-ferential algebra and H = (A ⊗K A)0 is the associated Hopf algebra, which is called the PV Hopf algebra for L/K. We can reprove 2.1 and 2.3 in the C-ferential context, so that the next two results follow. Proposition 7.4. The principal C-ferential algebra for a PV extension of C-ferential ﬁelds is simple. Theorem 7.5 (Galois correspondence I). Let (L/K, A, H ) be a PV extension of C-ferential ﬁelds. (1) There is a 1–1 correspondence, given by (2.7), between the Hopf ideals I in H and the intermediate C-ferential ﬁelds M in L/K. (2) If I ↔ M under the correspondence, (L/M, AM, H/I) is a PV extension. 8. Galois correspondence II and Characterization Throughout this section, we assume the following: Assumption 8.1. C is pointed irreducible, or in other words, the coradical of C equals R1C . This holds for the C in 6.4 and 6.5. The assumption ensures the following. Lemma 8.2 (Sweedler [16, Lemma 9.2.3]). Let A be an algebra. An element f in the algebra Hom(C, A) is invertible if and only if f(1C ) is invertible in A. The Extension Lemma 1.2 is generalized by the following. Proposition 8.3. Let A be a C-ferential algebra, and let T ⊂ A be a multiplicative subset. Then the C-ferential algebra structure on A uniquely extends via the canonical map A → T −1 A to such a structure on the localization T −1 A. Proof. Quite generally, if A is an algebra, the C-ferential algebra structures on A are in 1–1 correspondence with the sections A → Hom(C, A) of the algebra epimorphism

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Hom(C, A) A, f → f(1C ). We see from 8.2 that the composite ρ

cano

→ Hom(C, A) −−→ Hom(C, T −1 A), A− where ρ denotes the algebra map a → (c → ca) corresponding to the given C-ferential algebra structure, sends each t (∈ T ) to an invertible map in Hom(C, T −1 A) with value t/1 at 1C . Therefore, it uniquely extends to an algebra map T −1 A → Hom(C, T −1 A), which, as is easily checked, is the desired section. Given an extension L/K of C-ferential ﬁelds, an element x in L is said to be primitive (resp., exponential) over K, if cx ∈ K for all c ∈ C+ [resp., if it is nonzero and (cx)x−1 ∈ K for all c ∈ C]. By using the proposition stated earlier, we can restate word for word Examples 1.15, and therefore reprove the results in Section 5 (especially Theorem 5.8), all in the C-ferential context. Theorem 8.4 (Galois correspondence II). Let (L/K, A, H ) be a PV extension of C-ferential ﬁelds. Then, there is a 1–1 correspondence between the Hopf subalgebras J in H and the intermediate PV extensions M/K in L/K, in which to each M/K its PV Hopf algebra is assigned. Proof. Note from the proof of 8.3 that if A1 ⊂ L is a C-ferential subalgebra, the quotient ﬁeld L1 = Q(A1 ) of A1 (realized in L) is C-stable. Then, one sees that the proof of 2.8 is valid; see especially the sentence following (2.9). Just as in 2.11 and 2.13, the results 7.5 and 8.4 can be interpreted in terms of group schemes. We can restate Example 2.10 in the C-ferential context. Assumption 8.1 is equivalent to the property that the bialgebra T(C+ ) is pointed irreducible as a coalgebra. It follows that T(C+ ) is necessarily a Hopf algebra. It is easy to see the following, which generalizes 3.3. Lemma 8.5. Let A be a C-ferential algebra. Let V and W be a C-ferential A-modules [or left A#T(C+ )-modules]. Then the A-module HomA (V, W ) of A-linear maps V → W is a C-ferential A-module with respect to the C-action (8.6) (cf )(v) = c1 f(S(c2 )v) [c ∈ C, v ∈ V , f ∈ HomA (V, W )], where S denotes the antipode of T(C+ ). We have HomA (V, W )0 = HomA,C (V, W ), which denotes the A0 -module of C-ferential A-linear maps V → W . Let L/K be an extension of C-ferential ﬁelds, and let V be a C-ferential K-module. Let us write LV := HomK,C (V, L).

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By a slight modiﬁcation of 8.5, HomK (V, L) is a C-ferential L-module with HomK (V, L)0 = LV . By 6.7, we have a natural injection L ⊗L0 LV → HomK (V, L) in L,C M so that dimL0 LV ≤ dimK V . Deﬁnition 8.7. L is called a splitting ﬁeld for V if the following mutually equivalent conditions are satisﬁed (see 3.16): (i) There exists an injection L ⊗K V → λ Lλ in L,C M, where λ Lλ is the direct product of copies Lλ of L. (ii) L ⊗L0 LV → HomK (V, L) has a dense image. If dimK V = n < ∞, Conditions (i) and (ii) are equivalent to each of the following (see 3.9): (iii) L ⊗K V Ln in L,C M. (iv) L ⊗L0 LV − → HomK (V, L). V (v) dimL0 L = n. For a subset Z = {z1 , z2 , . . . } of L, let KZ = Kz1 , z2 , . . . denote the smallest intermediate C-ferential ﬁeld in L/K that contains Z. By 8.3, this equals the quotient ﬁeld of the K-subalgebra of L generated by T(C+ )Z. L/K is said to be ﬁnitely generated if L = Kz1 , . . . , zn for some ﬁnitely many elements z1 , . . . , zn in L. Let KV stand for K f ∈LV f(V ); this is the smallest intermediate C-ferential ﬁeld in L/K such that LV = KV V . A splitting C-ferential ﬁeld L for V is said to be minimal if L = KV . Theorem 8.8 (Characterization). Let L/K be an extension of C-ferential ﬁelds with the same constant ﬁeld k. (1) The following are equivalent: (i) L/K is a ﬁnitely generated PV extension. (ii) L is a minimal splitting ﬁeld for some C-ferential K-module of ﬁnite K-dimension that is cyclic, regarded as a K#T(C+ )-module. (iii) L is a minimal splitting ﬁeld for some C-ferential K-module V of ﬁnite K-dimension. (iv) There exists a GLn -primitive matrix X = (xij ) ∈ GLn (L) for some n such that L = K(xij ); by saying that X is GLn -primitive, we mean that for every c ∈ C, (cX)X−1 ∈ Mn (K), where cX = (cxij ). (2) The following are equivalent: (i) L/K is a PV extension. (ii) L is a minimal splitting ﬁeld for some C-ferential K-module that is locally ﬁnite in the sense that it is a sum of C-ferential K-submodules of ﬁnite K-dimension. Proof. (1) This is a modiﬁcation of the proof of 3.11. For (i) ⇒ (ii), let u1 , . . . , un and u = (u1 , . . . , un ) be just as in that proof. By 6.11, we can choose h1 , . . . , hn in

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T(C+ ) such that the matrix (hi uj ) in Mn (L) is invertible. Replace the W and the u(n) in (3.13) with W = (hi uj ) and hu = (hu1 , . . . , hun ) (h ∈ T(C+ )), respectively. Then, we see that (hu)W −1 ∈ Kn . This means that the cyclic Cferential K-submodule, say V , of Ln , which is generated by u, is K-spanned by the L-basis h1 u, . . . , hn u of Ln , whence L ⊗K V = Ln . Note in addition that KV = Ku1 , . . . , un , since LV is k-spanned by the n projections. For (iii) ⇒ (iv), we need to choose a K-basis v1 , . . . , vn of V , and replace (3.8) with ⎛ ⎞ ⎞ ⎛ cv1 v1 ⎜ .. ⎟ ⎜ . ⎟ ⎝ . ⎠ = (aij (c))⎝ .. ⎠ (c ∈ K, aij (c) ∈ K). cvn

vn

The xi in (3.14) should be replaced by xi = t (fi (v1 ), . . . , fi (vn )),

(8.9)

where f1 , . . . , fn constitute a k-basis of LV . We see that X = (x1 , . . . , xn )

(8.10)

is the desired GLn -primitive matrix. The further modiﬁcations should be obvious. (2) For this, the proof of 3.17 can be copied since 3.15 is directly generalizable in the C-ferential context. To state a result without counterpart in Part I, let K be a C-ferential ﬁeld with k = K0 . There is the k-Abelian symmetric tensor category, (K,C M, ⊗K , K). Let V ∈ ∗ K,C M be of ﬁnite K-dimension. The K-dual V := Hom K (V, K) is in K,C M; see 8.6. Therefore, the full tensor subcategory, K,C Mﬁn , consisting of all ﬁnite K-dimensional objects in K,C M is rigid. Let {{V }} denote the Abelian, rigid full tensor subcategory of K,C Mﬁn generated by V , that is, the smallest full subcategory containing V that is closed under subquotients, ﬁnite direct sums, tensor products, and duals. Thus, an object in {{V }} is precisely a subquotient of some W1 ⊕ · · · ⊕ Wr , where each Wi is the tensor product of copies of V, V ∗ . Theorem 8.11 (cf. [4, Theorem 2.33]). Let (L/K, A, H ) be a ﬁnitely generated PV extension of C-ferential ﬁelds. By 8.8 (1), we have an object V ∈ K,C Mﬁn for which L is a minimal splitting ﬁeld. (1) Let W ∈ {{V }}. Regard A ⊗K W as a right H-comodule with respect to the structure induced by A = (A, θ); see 7.2 (2). Then, (A ⊗K W )0 is an H-subcomodule with dimk (A ⊗K W )0 = dimK W . (2) W → (A ⊗K W )0 gives a k-linear equivalence {{V }} ≈ MH ﬁn

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H of symmetric tensor categories, where MH ﬁn = (Mﬁn , ⊗k , k) denotes the rigid symmetric tensor category of ﬁnite-dimensional right H-comodules; note that this is isomorphic to the category of the same kind RepG(L/K) , which consists of the ﬁnite-dimensional k-linear representations of G(L/K) = Sp H, the PV group scheme of L/K.

Proof. Let (C MH , ⊗k , k) denote the symmetric tensor category consisting of those C-ferential k-modules N, which have a C-ferential, right H-comodule structure N → N ⊗k H. Since A is an algebra in C MH , we have the symmetric tensor category (A,C MH , ⊗A , A) of A-modules in C MH . Since A/K is H-Galois by 7.2 (2), it follows from [22, Theorem 2.11] (or see [14, Theorem 8.5.6, (1) ⇒ (3)]) that W → A ⊗K W gives an equivalence K,C M ≈ A,C MH ; this is a k-linear equivalence of symmetric tensor categories. Deﬁne k-linear functors,

→ (MH , ⊗k , k) − MH (≈ K,C M), ← A,C

(N ) = N0 ,

(U ) = A ⊗k U

(codiagonal H-coaction),

where the category of the left-hand side is that of right H-comodules. Obviously, is a symmetric tensor functor. Since A0 = k, ◦ id. By 6.7 and 7.4, the natural morphism in A,C MH μN : ◦ (N ) = A ⊗k N0 → N,

μN (a ⊗k n) = an,

is injective. Let N denote the full subcategory in A,C MH consisting of those N for which μN is bijective. Since (U ) ∈ N , N is closed under tensor products and gives an equivalence MH ≈ N of symmetric tensor categories. Part 2 of the theorem will follow if we prove (a) A ⊗K V ∈ N ,

(b) (A ⊗K V )0 = (A ⊗K V ) ∈ MH ﬁn , and (c) (A ⊗K V )0 generates MH ﬁn . Choose a K-basis v1 , . . . , vn of V and set v = (v1 , . . . , vn ). Let xi = t (fi (v1 ), . . . , fi (vn )) (1 ≤ i ≤ n) be as in (8.9) and set X = (xij ) = (x1 , . . . , xn ) as in 1 ]. We see that 1 ⊗K v → t X, or 1 ⊗ (8.10). Recall from 1.16 that A = K[xij , |X|

vi → (f1 (vi ), . . . , fn (vi )) (1 ≤ i ≤ n), gives an isomorphism A ⊗K V − → An in A,C M, which implies (a) and (b). It also follows that (A ⊗K V )0 has the entries in (1 ⊗K v)(t X−1 ⊗K 1) as a k-basis of it. Let B = (bij ) = (X−1 ⊗K 1)(1 ⊗K X); see 1.16. Since θ(t X−1 ) = (t X−1 ⊗k 1)(1 ⊗k t B−1 ) in Mn (A ⊗k H ), we see that the coefﬁcient space of the right H-comodule (A ⊗K V )0 is k-spanned by S(bij ), the entries of t B−1 ; this implies (c). If W ∈ {{V }}, then A ⊗K W ∈ N . Hence, dimk (A ⊗K W )0 equals the free A-rank of A ⊗K W or dimK W ; this proves Part 1.

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9. Uniqueness and existence of minimal splitting ﬁelds In general, a coalgebra D is said to be Birkhoff–Witt if it is isomorphic to the cofree pointed irreducible cocommutative coalgebra B(U ) on some vector space U; see [16, p. 261]. Such a D is necessarily pointed irreducible and cocommutative. Suppose that D is an irreducible cocommutative bialgebra, which is necessarily a Hopf algebra. If ch R = 0, then D is necessarily Birkhoff–Witt. If ch R = p > 0, then D is Birkhoff– Witt if and only if the Verschiebung map VD : D → R1/p ⊗ D is surjective [23, Corollary 4.2.7 (b)]. Recall that the map VD is deﬁned for every cocommutative coalgebra D, as the dual notion of the Frobenius map for commutative algebras. Throughout this section we assume the following: Assumption 9.1. T(C+ ) is Birkhoff-Witt. If ch R = 0, this is equivalent to Assumption 8.1 that C is pointed irreducible. If ch R = p > 0, the assumption stated afore holds if C is pointed irreducible and if VC is surjective. These last two conditions are true if C is spanned by divided power sequences. Therefore, such C as in Example 6.5 (2) satisfy the assumption in arbitrary characteristic, while those C in Examples 6.4 and 6.5 (1) do so only when ch R = 0; see the following example below. Let A be a C-ferential algebra. Recall from 6.8 that Hom(T(C+ ), A) is a C-ferential A-algebra endowed with the C-ferential algebra map ρ˜ : A → Hom(T(C+ ), A) given by (6.9). Example 9.2. Suppose ch R = 0 and C = R1C ⊕Rd, as in 6.4. Then, T(C+ ) = R[d], 2 3 the polynomial Hopf algebra, in which 1, d, d2! , d3! , . . . form a divided power sequence; see 6.5 (2). We can identify Hom(T(C+ ), A) = A[[t]], by means of the pairing d i , t j = δij i! The map ρ˜ above generalizes the map ρ : A → A[[t]] deﬁned in Lemma 4.1 (1). Lemma 4.1 (4) generalizes to the following. Lemma 9.3. If a C-ferential algebra A is simple, it is an integral domain. Proof. Just as in the proof of 4.1 (4), it sufﬁces to prove the following in the order given: + (a) An ideal I of A is C-ferential if and √ only if I = ρ˜ −1 (Hom(T(C √ ), I)). −1 + (b) If I ⊂ A is a C-ferential ideal, then I = ρ˜ (Hom(T(C ), I)). (c) If P ⊂ A is a prime ideal, then Hom(T(C+ ), P) is prime.

It is easy to prove (a).Assumption 9.1 implies that Hom(T(C+ ), A) is a projective limit of A-algebras, A[[t1 , . . . , tr ]], of formal power series. Therefore, Hom(T(C+ ), A) is reduced (resp., an integral domain) if A is. This implies (b) and (c). Using 8.3 and 9.3, we can now reprove Lemma 4.2 in the C-ferential context.

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Theorem 9.4 (Unique existence). Suppose that K is a C-ferential ﬁeld whose constant ﬁeld K0 is algebraically closed. Let V be a C-ferential K-module of ﬁnite K-dimension. There exists a minimal splitting ﬁeld L for V with L0 = K0 so that L/K is necessarily a ﬁnitely generated PV extension of C-ferential ﬁelds. Such an L is unique up to isomorphism of C-ferential K-algebras. Proof. Modify the proof of 4.3, by using the just reproved lemma together with 7.4 and 8.3. We end this part with an example in positive characteristic. Example 9.5. Let C be as in Example 6.5 (2), but write k for R. Recall that this satisﬁes Assumption 9.1. Both the algebra k[[t]] of formal power series and its quotient ﬁeld k((t)) are C-ferential algebra with respect to the action di t j = ji t j−i (i ≤ j), di t j = 0 (i > j), so that k[[t]]0 = k((t)) 0 = k. Note that C is a commutative Hopf algebra with respect to the product di dj = i+j i di+j and that the C-action above coincides with the natural action on the dual algebra C∗ = k[[t]]. Suppose ch k = p > 0. In the dual Hopf algebra C◦ (⊂ k[[t]]) of C [16, Section 6.2], pick the primitive t, and choose another pi primitive f = 0; this must be an element of the form f = ∞ i=0 ai t (ai ∈ k) and all these quality. Now take the (Hopf) subalgebra, J := k[t, f ]. The Hopf subalgebra J1 := k[t]∩k[f ] in the polynomial Hopf algebra k[f ] is generated by a primitive, say ν(f ). Since k[t] ⊗J1 k[f ] = J, we see J/(t) = k[f ]/(ν(f )). Let H = k[f ]/(ν(f )), K = k(t), A = K⊗k[t] J = K[f ], and L = Q(A) = k(t, f ). The natural isomorphism J ⊗k[t] J − → J ⊗ H, x ⊗ y → xy1 ⊗ y¯ 2 is C-ferential, where C acts on the factor J in J ⊗H. By applying K⊗k[t] , we have a C-ferential isomorphism A⊗K A − → A⊗H ∞ pi so that (L/K, A, H ) is PV. For example, if f = i=0 t , then ν(f ) = f − f p (= t) so that L = A = K[f ]/(f p − f + t) and H = k[f ]/(f p − f ) k(Z/(p))∗ . If i pp , then ν(f ) = 0 so that A = K[f ], H = k[f ] are polynomial algebras. f = ∞ i=0 t Part III: Uniﬁed PV theory In this part, we work over a ﬁxed ground ﬁeld R, keeping the conventions stated at the beginning of Part II, but C will be replaced by a special kind of D. 10. What’s the difference in the difference case To move from the differential world to the difference world, one has only to exchange “derivation” to “automorphism.” A difference ﬁeld is thus a ﬁeld K together with a single (speciﬁed) automorphism; this is what is called an inversive difference ﬁeld by Cohn [24]. A homogeneous linear difference equation with coefﬁcients in K is an equation of the form φn y + a1 φn−1 y + · · · + an−1 y = 0

(ai ∈ K),

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where φ is understood to be an automorphism. This notion can be generalized to difference systems (or even to difference modules), just as linear differential equations are generalized to differential systems (and differential modules). Linear recurrence equations with coefﬁcients, say, in the complex number ﬁeld C are regarded as difference equations over the difference ﬁeld (C, id C ), whose solutions lie in the C-algebra CZ = Map(Z, C) of C-valued maps on Z; this is a difference ring with constants C, with respect to the shift operator φ{αn }n = {αn+1 }n , where {αn }n ∈ CZ . Example 10.1. Consider the Fibonacci recurrence equation αn+2 − αn+1 − αn = 0.

(10.2)

This is regarded as a difference equation by setting φ(αn ) = αn+1 .√As is wellknown, its solutions in CZ form a C-vector space with basis α = {( 1+2 5 )n }n , β = {(− 2√ )n }n . Let A be the difference C-subalgebra of CZ generated by α, β. It 1+ 5

contains the nontrivial idempotent 12 (αβ + 1) = {. . . , 1, 0, 1, 0, . . . }, as though we could say that (10.2) splits in A. This suggests that the difference ﬁelds could not be adequate to work with when we develop difference PV theory. We remark that α → (t, 1t ), β → ( 1t , −t) gives an embedding of A into the difference C-algebra C(t) × C(t) √

with the automorphism φ(t, 0) = (0, 1+2t 5 ), φ(0, t) = ( 2√ , 0). Precisely as in (1+ 5)t this example, the product of copies of ﬁelds equipped with an automorphism that acts transitively on the primitive idempotents will turn out to be a desired object. To extend the framework so as to include difference PV theory as well, we have to replace, on the one hand, the C [or rather T(C+ )] in Part II by some D that contains grouplikes, and on the other hand, replace dif- or C-ferential ﬁelds by some generalized object. These are precisely the AS D-module algebras as will be deﬁned in 11.6, while D is as follows.

Notation 10.3. D denotes a pointed cocommutative Hopf algebra, which may not be commutative. Let G = G(D) denote the group of all grouplikes in D, and let D1 denote the largest irreducible Hopf subalgebra in D. The multiplication d ⊗ g → dg → D of coalgebras. Since D1 is stable under gives an isomorphism D1 ⊗ RG − G-conjugation, we have D = D1 #RG. If the C as in Part II is pointed, then T(C+ ) can be D. We will later impose some additional assumption on D1 ; see 11.4. The left D-modules form a symmetric tensor category, (D M, ⊗, R). Given V ∈ we deﬁne the subspace of D-invariants by

D M,

V D = {v ∈ V | dv = ε(d)v

(d ∈ D)}.

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This notation is used, instead of V0 , since D will possibly change in the sequel. Recall that a D-module algebra is a commutative algebra in (D M, ⊗, R). The result 8.3 generalizes as follows. Proposition 10.4. Let A be a D-module algebra, and let T ⊂ A be a G-stable multiplicative set. Then the D-module algebra structure on A uniquely extends to such a structure on T −1 A. Proof. By [14, Lemma 5.2.10] due to Takeuchi, the algebra map ρ : A → Hom(D, A), a → (d → da) uniquely extends to an algebra map ρ˜ : T −1 A → Hom(D, T −1 A), since in Hom(D, T −1 A), each ρ(t) (t ∈ T ) is invertible on the coradical RG of D. The associated D-action d(a/t) = ρ(a/t)(d) ˜ makes T −1 A into −1 a D-module, since the two maps D ⊗ D → T A given by c ⊗ d → (cd)(1/t) and c ⊗ d → c(d(1/t)) coincide, both being inverse to c ⊗ d → cdt. A D-module algebra A = 0 is simple if it includes no nontrivial D-stable ideal or in other words if it is simple as an A#D-module. By modifying 6.7 in the obvious way we have the simplicity criterion of a D-module algebra; see also Proposition 12.5. We have especially that if A is simple, AD is a ﬁeld. 11. AS D-module algebras Let G1 ⊂ G be a subgroup of ﬁnite index. Deﬁne a Hopf subalgebra of D = D1 #RG by D(G1 ) = D1 #RG1 . We will identify the ﬁnite set G/G1 of left cosets with some ﬁxed system of their representatives in G. By convention, we choose 1 as one representative. Set D1 = D(G1 ), and let V be a left D1 -module. We denote the left D-module D ⊗D1 V by G1 (V ) or by (V ). This is identiﬁed as (V ) = g ⊗ V, g∈G/G1

where by g ∈ G/G1 , we mean that g runs over the ﬁxed representatives for G/G1 . D acts the right hand side so that if d ∈ D1 , d(g ⊗ v) = g ⊗ g−1 dgv

(v ∈ V )

and if h ∈ G, h(g ⊗ v) = g ⊗ h v

(v ∈ V ),

where g ∈ G/G1 , h ∈ G1 , such that hg = g h .

→ (V )D is given by v → Lemma 11.1. A natural isomorphism V D1 −

g∈G/G1 g⊗v.

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Keep the notation as above. Let A1 be a D1 -module algebra. Then, (A1 ) is a D-module algebra with respect to the “component-wise” product (g ⊗ a)(g ⊗ a ) = δg,g g ⊗ aa

(a, a ∈ A1 ).

Thus the g ⊗ 1 (g ∈ G/G1 ) are orthogonal idempotents in (A1 ) whose sum is 1. Set A = (A1 ). Note that if V is a left A1 #D1 -module, then (V ) is a left A#D-module with respect to the component-wise A-action. Proposition 11.2. G1 : A1 #D1 M → A#D M is a category equivalence. The A1 -bimodules in (D1 M, ⊗, R) form a tensor category, (A1 (D1 M)A1 , ⊗A1 , A1 ). Similarly, we have (A (D M)A , ⊗A , A). Proposition 11.3. G1 : (A1 (D1 M)A1 , ⊗A1 , A1 ) → (A (D M)A , ⊗A , A) gives a tensor equivalence onto the full tensor subcategory of A (D M)A , which consists of the objects M such that (g ⊗ 1)m = m(g ⊗ 1)(m ∈ M) for every (or equivalently, an arbitrarily chosen) g ∈ G/G1 . Subsequently, we assume the following. Assumption 11.4. D1 is Birkhoff-Witt. This is equivalent to saying that D is smooth as a cocommutative coalgebra. Since 11.4 is the same as Assumption 9.1 if D = T(C+ ), the results in Section 9 can be recovered from the forthcoming results by taking D = D1 = T(C+ ). Let K be a D-module algebra. Suppose that K is Noetherian as a ring. Let (K) denote the (ﬁnite) set of all minimal prime ideals in K. Then G acts on (K). For P ∈ (K), let GP denote the subgroup of stabilizers of P. Set G(K) = P∈(K) GP ; this is a normal subgroup in G of ﬁnite index. Proposition 11.5. Suppose in addition that K is simple. (1) The action of G on (K) is transitive so that the subgroups GP [P ∈ (K)] are conjugate to each other. (2) Every P ∈ (K) is D1 -stable so that K/P is a D(GP )-module domain. This is simple as a D(G(K) )-module algebra. (3) Let P ∈ (K) and set K1 = K/P. Then, we have a natural isomorphism K GP (K1 ) of D-module algebras. Proof. (2) Just as in the proof of 9.3, we see that Hom(D1 , P) is a prime in Hom(D1 , K), and Q := ρ−1 (Hom(D1 , P)) is a D1 -stable prime in K, which is included in P. By its minimality, P must equal Q and be D1 -stable. For the second assertion, let P ⊂ J K be a D(G(K) )-stable ideal. Then, g∈G/G(K) gJ is D-stable and hence is zero. There exists a g such that gJ ⊂ P; this implies P ⊂ J ⊂ g−1 P, whence P = J by the minimality of g−1 P.

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(1) If P ∈ (K), then G} = (K), as desired.

g∈G gP,

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being D-stable, must be zero. Hence, {gP | g ∈

→ (K). If g = g in G/GP , (3) By (1), g → gP gives a bijection G/GP − → g∈G/GP K/gP = g∈G/GP g ⊗ K1 = then gP + g P = K by (2). Hence, K − GP (K1 ), as desired. A ring K is said to be total if every nonzero-divisor in K is invertible. In Part III, we let Q(K) stand for the total quotient ring of K; this is by deﬁnition the localization of K by the multiplicative set of all nonzero-divisors. Hence, K is total if and only if K = Q(K). Lemma-Deﬁnition 11.6. Let K = 0 be a D-module algebra. The following are equivalent: (i) K is Artinian and simple (as a D-module algebra); (ii) K is Noetherian, total, and simple. If these conditions are satisﬁed, K is said to be AS. In this case, every K#D-module is free as a K-module. Proof. Suppose that K is Noetherian and simple. By 11.5, K = P∈(K) K/P. Both (i) and (ii) are equivalent to the statement that each K/P is a ﬁeld. Choose P ∈ (K) and set K1 = K/P. By 11.3 and 11.5 (3), every K#D-module is of the form GP (V ) = g∈G/GP g ⊗ V , where V is a K1 #D(GP )-module; this implies the last assertion. Remark 11.7. A D-module that is a ﬁeld is obviously AS; it will be called a D-module ﬁeld. A C-ferential ﬁeld as deﬁned in Part II is precisely a T(C+ )-module ﬁeld, where for Assumption 11.4, we need to assume 9.1. On the other hand, a D-module ﬁeld is necessarily a D-ferential ﬁeld. Lemma 11.8. Let G1 ⊂ G be a subgroup of ﬁnite index. A D(G1 )-module algebra K1 is AS if and only if the D-module algebra G1 (K1 ) is AS. Lemma 11.9. Let L be an AS D-module algebra, and let A ⊂ L be a D-module subalgebra. (1) Every nonzero-divisor of A is a nonzero-divisor of L so that the total quotient ring Q(A) is realized in L. (2) This Q(A) is a D-module subalgebra of L, which is AS. Proof. (1) Given 0 = x ∈ L, deﬁne its support by Supp(x) = {P ∈ (L) | x ∈ P}. Note that x is a nonzero-divisor of L if and only if Supp(x) = (L). Let x be a nonzero-divisor of A whose support is minimal among the nonzero-divisors of A. If

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Supp(x) = (L), then by 11.5 (1) and the minimality, there exists a g ∈ G such that Supp(x) ∩ Supp(gx) = ∅; this implies x(gx) = 0, a contradiction. (2) By 10.4, Q(A) is a D-module subalgebra of L. It is necessarily AS, since a total D-module subalgebra of an AS D-module algebra is AS, as is proved in [11, Lemma 2.8]. 12. PV extensions of AS D-module algebras Let L ⊃ K be an inclusion of AS D-module algebras; it is also referred to as an extension L/K. There is the corresponding ﬁeld extension LD/KD . Deﬁnition 12.1. We say that L/K is a PV extension if (i) LD = KD , which is a ﬁeld and will be denoted by k, and (ii) there exists an intermediate D-module algebra L ⊃ A ⊃ K such that (a) L = Q(A) [see 11.9 (2)], (b) A ⊗K A is generated by (A ⊗K A)D as a left (or equivalently right) A-module. Such an A as above is called a principal D-module algebra for L/K. Let L/K and A be as above. Set H = (A ⊗K A)D ; this is a k-subalgebra of A ⊗K A. By using the simplicity of L, it follows that μ : A ⊗k H → A ⊗K A, μ(a ⊗k h) = (a ⊗K 1)·h is an isomorphism. Then, we can now reprove Lemmas 7.2 and 7.3 in the present context. Thus, H has a natural structure of commutative k-Hopf algebra; it is called the PV Hopf algebra for L/K. A principal D-module algebra A for L/K is unique; it has a natural structure θ : A → A ⊗k H of a right H-comodule algebra for which A/K is an H-Galois extension. We say that (L/K, A, H ) is a PV extension, indicating A, H as well. Example 12.2. Let G1 ⊂ G be a normal subgroup of ﬁnite index. Let K be a D-module ﬁeld. Regarding this as a D(G1 )-module ﬁeld, deﬁne L := G1 (K). There is an inclusion of D-module algebras, K → L = g ⊗ K, x → g ⊗ g−1 x. g∈G/G1

g

If KD(G1 ) = KD , then KD = LD (=: k) by 11.1. Moreover, (L/K, L, H ) is a PV extension, where H = k(G/G1 )∗ . In fact, we see that the elements (h ⊗ 1) ⊗K (hg ⊗ 1) (g ∈ G/G1 ) eg := h∈G/G1

in L ⊗K L are D-invariant and behave as the dual basis of the group elements g , S(eg ) = eg−1 . The in k(G/G1 ). Thus, (eg ) = h egh−1 ⊗k eh , ε(eg ) = δ1,g H-comodule structure θ : L → L⊗k H is given by θ(h⊗x) = g (hg−1 ⊗gx)⊗k eg .

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Lemma 12.3. Let G1 ⊂ G be a subgroup of ﬁnite index, and write = G1 . Let L1 /K1 be an extension of AS D(G1 )-module algebras; (L1 )/(K1 ) is then also an extension of AS D-module algebras. (L1 /K1 , A1 , H ) is a PV extension if and only if ((L1 )/(K1 ), (A1 ), H ) is a PV extension. Proof. The natural coalgebra isomorphism (A1 ⊗K1 A1 ) (A1 ) ⊗(K1 ) (A1 ) (see 11.3) together with 11.1 proves the lemma. Remark 12.4. Let L/K be an extension of AS D-module algebras. Choose p ∈ (K), and let P1 , . . . , Pr be those primes in (L), which lie over p. Deﬁne K1 = K/p, L1 = ri=1 L/Pi . Then, we have an extension L1 /K1 of AS D(Gp )-module algebras such that the induced (L1 )/(K1 ) is identiﬁed with L/K. Replacing L/K with L1 /K1 , we can often suppose that K is a ﬁeld when discussing PV extensions; see 12.3. To give a general result, which holds in an arbitrary Abelian category A, ﬁx an object X in A, and set E = A(X, X), the endomorphism ring of X. Note that for every object Y in A, A(X, Y ) is naturally a right E-module. Proposition 12.5 (Simplicity criterion). X is simple (i.e., is nonzero and includes no nontrivial subobject) if and only if (a) E is a division ring, and (b) for every object Y in A, the morphism (f1 , . . . , fn ) : Xn = X ⊕ · · · ⊕ X → Y given by ﬁnitely many, E-linearly independent morphisms f1 , . . . , fn in A(X, Y) is a monomorphism. Proof. “If”. Let Z X be a proper subobject. The quotient map f in A(X, X/Z), being nonzero, is E-linearly independent by (a). By (b), f : X → X/Z is monic, and so Z = 0. “Only if”. The argument that proves Schur’s lemma shows (a). We prove (b) by induction on the number n. The proof in case n = 1 is obvious. Let n > 1, and assume contrary to (b) that (f1 , . . . , fn ) is not monic, while f1 , . . . , fn are E-linearly independent. Set g = (f1 , . . . , fn−1 ); this g as well as fn is monic by the induction hypothesis. Let Y ⏐fn ⏐

⏐ ⏐

Z

g

⏐ ⏐

X n−1 ⏐ ⏐

X

be a pull-back diagram, which consists of monomorphisms. By the assumption above, → X by the simplicity of X. The induced morphism X Z → Z = 0, and so Z −

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Xn−1 is of the form t (a1 , . . . , an−1 ) with ai ∈ E. This implies fn = contradicting the E-linear independence.

n−1 i=1

fi a i ,

Note that Propositions 1.3, 3.2, and 6.7 can all be obtained as special cases of the proposition just proved. Theorem 12.6 (Galois correspondence). Let (L/K, A, H ) be a PV extension of AS D-module algebras. (1) There is a 1–1 correspondence, given by (2.7), between the Hopf ideals I in H and the intermediate AS D-module algebras M in L/K. (2) If I ↔ M under the correspondence, (L/M, AM, H/I ) is a PV extension. (3) If I ↔ M under the correspondence, the Hopf ideal I is normal if and only if M/K is a PV extension. By [18, Theorem 4.3], this gives rise to a 1–1 correspondence between the Hopf subalgebras J in H and the intermediate PV extensions M/K in L/K, in which to each M/K its PV Hopf algebra is assigned. Proof. Once Part 1 is proved, Parts 2 and 3 will follow in the same way as for 2.6 (2) and 2.8. For Part 1, we need to prove 2.1 and 2.3 in the present, generalized situation. For 2.3, this is easy. For 2.1, suppose that M is given. Since L, being an M#D-module, is M-free, M can be recovered from the corresponding D-stable coideal Ker(L ⊗K L → L ⊗M L) in the coalgebra L ⊗K L in (L (D M)L , ⊗L , L). Conversely, let a ⊂ L ⊗K L be a D-stable coideal. Deﬁne M = Ma by 2.2; this is obviously an intermediate D-module algebra, which is seen to be total, and so AS by 11.9 (2). Set C = L ⊗K L/a. One sees that the quotient map L ⊗K L C factors through a coalgebra surjection γ : L ⊗M L C. We need to prove that γ is injective. By 11.3, we may suppose that M is a ﬁeld. To apply 12.5, regard C just as an L-coring or a coalgebra in (L ML , ⊗L , L), and suppose that A is the category of right C-comodules; an object Y in A is thus a right L-module together with an L-linear comodule structure map Y → Y ⊗L C. Take L as the X in 12.5; it has the natural C-comodule structure λ : L → L ⊗L C = C, λ(x) ≡ 1 ⊗K x mod a. Since E = A(L, L) M, A(L, C) L, Condition (b) in 12.5, applied to Y = C, is equivalent to injectivity of γ. Therefore, it sufﬁces to verify the assumption of 12.5 that L is simple in A. A subobject of L is of the form eL, where e is an idempotent. Since λ is D-linear, g(eL), where g ∈ G, is also a subobject so that L is semisimple by 11.5 (1). However, L must be simple, since the endomorphism ring E is a ﬁeld. Just as in the previous parts I and II, the result above can be interpreted in terms of group schemes. Corollary 12.7. Let (L/K, A, H ) be a PV extension of AS D-module algebras. (1) A is simple as a D-module algebra. (2) A contains all primitive idempotents in L so that A = P∈(L) A/P ∩ A.

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Proof. (1) See the proof of 2.5. (2) Since A → L is a localization, we have (A) ⊃ (L). We need to prove that if P = Q in (L), the sum J := P ∩A+Q∩A equals A. If on the contrary J A, one sees from (1), as in the proof of 11.5 (2), that J = P ∩ A = Q ∩ A, which implies P = Q. Proposition 12.8. Let (L/K, A, H ) be a PV extension of AS D-module algebras. Choose P ∈ (L) arbitrarily, and write = GP . Set p = P ∩ K (∈ (K)), and deﬁne K1 = K/p, A1 = A/P ∩ A, L1 = L/P. Then (1) A (A1 ). ˆ which is K-spanned by (2) (K1 ) identiﬁes with the K-subalgebra of L, say K, the primitive idempotents in L. ¯ := H/I ) is a PV extension of D(GP )-module ﬁelds, where (3) (L1 /K1 , A1 , H ˆ I = H ∩ Ker(L ⊗K L → L ⊗Kˆ L), the Hopf ideal corresponding to K. (4) The subalgebra of H B := {h ∈ H | (h) − h ⊗ 1 ∈ H ⊗ I} ¯ is a separable k-algebra of dimension [Gp : GP ]. There is a right H-colinear ¯ B-algebra isomorphism H B ⊗k H. (5) The subgroup GP ⊂ Gp is normal if and only if B splits in the sense of B k × · · · × k. In this case, B ⊂ H is a Hopf subalgebra isomorphic to k(Gp /GP )∗ so that there is a short exact sequence of k-Hopf algebras, ¯ k(Gp /GP )∗ H H. Proof. (1) This follows by 12.7 (2). (2) This is easy. (3) By 12.6 (2), we have ˆ A, H) ¯ = ((L1 )/(K1 ), (A1 ), H). ¯ Part 3 now follows a PV extension, (L/K, from 12.3. (4) and (5). By 12.3, we may suppose that K is a ﬁeld, and so p = 0, G = Gp . ˆ → A ⊗K A → The obvious equalizer diagram 0 → A ⊗K K → A ⊗K A ⊗Kˆ A of D-module algebras is naturally identiﬁed with 0 → A ⊗k B → A ⊗k H → → A ⊗k ¯ Induced is an isomorphism A ⊗k B A ⊗K K. ˆ By applying L1 ⊗A , we H ⊗k H. obtain ˆ = (L1 ) = L. L1 ⊗k B L1 ⊗K K (12.1) This implies the ﬁrst assertion of (4). For the second, note that σ : (A1 ⊗K A1 ) = A ⊗Kˆ A → A ⊗K A given by σ(g ⊗ (a ⊗K b)) = (g ⊗ a) ⊗K (g ⊗ b) (g ∈ G/GP ) is a ¯ D-linear, right H-colinear k-algebra splitting of A ⊗K A → A ⊗Kˆ A. The restriction ¯ → H to D-invariants is an H-colinear ¯ ¯ k-algebra splitting of H → H. σD : H ¯ → H, It follows from [25, Theorem 9] (or see [14, Theorem 7.2.2]) that B ⊗k H b ⊗ x → bσ D (x) is the desired isomorphism. ˆ If GP G, then by 12.2, K/K is a PV extension with PV Hopf algebra k(G/GP )∗ . ∗ We must have B = k(G/GP ) , which splits. If B splits, it follows from the isomorphism (12.1), which preserves the GP -action, that GP stabilizes each component in L, whence GP G. This proves the ﬁrst assertion of (5). The second one is easy.

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Theorem 12.10. Let L/K be an extension of AS D-module algebras. Choose an arbitrary P ∈ (L), and set p = P ∩ K (∈ (K)). Set K1 = K/p, L1 = L/P. Then L/K is a PV extension if (i) GP is normal in Gp , and (ii) the extension L1 /K1 of D(GP )-module ﬁelds is PV. The converse is true if the ﬁeld KD ( = LD ) is separably closed. Proof. If KD is separably closed and L/K is PV, then the separable algebra B as given in 12.8 (4) splits, which implies (i) above, by 12.8 (5). This together with 12.8 (3) proves the last assertion. ¯ For what remains, we can suppose that K is a ﬁeld. Suppose that (L1 /K1 , A1 , H) is PV. Let = GP , and deﬁne A = (A1 ). If L/K is PV, the principal D-module ¯ ¯ algebra with Halgebra must be A; see 12.8 (1). A ⊗K A is a right H-comodule ¯ coinvariants A ⊗K (K). We can deﬁne a D-linear, H-colinear k-algebra map σ : ¯ (A1 ⊗K A1 ) → A⊗K A as in the last proof. Induced is an H-colinear k-algebra map D D ¯ ¯ σ : H → (A ⊗K A) . Again by [25, Theorem 9], we have a D-linear, H-colinear ¯ − → A ⊗K A over A ⊗K (K). Hence, L/K is algebra isomorphism A ⊗K (K) ⊗k H PV if and only if the natural injection A⊗k (A⊗k (K))D → A⊗k (K) is surjective. Indeed, this is surjective if GP G, since then by 12.2, A ⊗k ((K) ⊗K (K))D → A ⊗k (K) is already surjective. There is an example [11, Example 3.16] of a PV extension of the form G1 (K)/K, where K is a D-module ﬁeld, and G1 ⊂ G is a non-normal subgroup of ﬁnite index. This shows that, in 12.10, the converse does not necessarily hold without the assumption of separable closedness. 13. Characterization and unique existence Let L/K be an extension of AS D-module algebras. Let V be a K#D-module, which is necessarily K-free; see 11.6. Its K-free rank will be denoted by rk K V and simply called the K-rank. By slightly modifying the argument in Section 8, we see that HomK (V, L) is naturally an L#D-module [see (8.6)], whose D-invariants equal the LD -vector space LV := HomK#D (V, L). There is a natural injection L ⊗LD LV → HomK (V, L) in L#D M, which necessarily splits L-linearly so that dimLD LV ≤ rk K V . Deﬁnition 13.1. Let V be a K#D-module. L is a splitting algebra for V if there exists an injection L⊗K V → λ Lλ in L#D M, where λ Lλ is the product of copies Lλ of L. If rk K V = n < ∞, the condition is equivalent to each of the following: → HomK (V, L); dimLD LV = n. L ⊗K V Ln in L#D M; L ⊗LD LV −

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For a subset Z = {z1 , z2 , . . . } of L, let KZ = Kz1 , z2 , . . . denote the smallest intermediate AS D-module algebra in L/K that includes Z. This coincides with the localization T −1 A of the K-subalgebra A ⊂ L generated by the D-submodule DZ, where T is the multiplicative set consisting of those elements x in A with full support, i.e., Supp(x) = (L); see the proof of 11.9. L/K is said to be ﬁnitely generated if L = Kz1 , . . . , zn for some ﬁnite elements z1 , . . . , zn in L. If L/K is PV, this, as will be seen from 13.3, is equivalent to the following: L is the total quotient ring of some ﬁnitely generated K-subalgebra in L. Let KV stand for K f ∈LV f(V ). A splitting algebra L for V is said to be minimal if L = KV . Lemma 13.2. Let G1 ⊂ G, K1 ⊂ L1 be as in 12.3. Write = G1 . Then, L1 is a (minimal) splitting algebra for a K1 #D(G1 )-module V1 if and only if (L1 ) is a (minimal) splitting algebra for the (K1 )#D-module (V1 ). Proof. This follows from 11.2; note that (K1 V1 ) = (K1 )(V1 ).

Theorem 13.3 (Characterization). Let L/K be an extension of AS D-module algebras such that LD = KD . (1) The following are equivalent: (i) L/K is a ﬁnitely generated PV extension. (ii) L is a minimal splitting algebra for some cyclic K#D-module of ﬁnite K-rank. (iii) L is a minimal splitting algebra for some K#D-module of ﬁnite K-rank. (iv) There is a GLn -primitive matrix X = (xij ) ∈ GLn (L) for some n such that L = Kxij ; by saying that X is GLn -primitive, we mean that for every d ∈ D, (dX)X−1 ∈ Mn (K), where dX = (dxij ). (2) The following are equivalent: (i) L/K is a PV extension. (ii) L is a minimal splitting algebra for some K#D-module that is a sum of K#D-submodules of ﬁnite K-rank. Proof. (1) We prove only the implication (i) ⇒ (ii). For the rest just adapt the proofs of 3.11 and 8.8. Suppose that (L/K, A, H ) is ﬁnitely generated PV. By 12.3 and 13.2, we can suppose that K is a ﬁeld. By 12.8 (3), we have a ﬁnitely generated ¯ of module ﬁelds over D1 = D(GP ) with P ∈ (L) PV extension (L1 /K1 , A1 , H) such that L = GP (L1 ), A = GP (A1 ). Write k = KD (= LD ). There exist ﬁnitely many elements u1 , . . . , un in A = (A, θ) such that they kspan an H-subcomodule and satisfy L = Ku1 , . . . , un . Take the element u = (u1 , . . . , un ) in An , and let V = (K#D)u, the cyclic K#D-module generated by u. Since L⊗K A L⊗k H, we see that L is a minimal splitting algebra for An and hence for V . It remains to prove dimK V < ∞. For this, it sufﬁces to prove that the natural image V(P), say, of V under the projection An → An1 has ﬁnite K-dimension since V → P∈(L) V(P). Let g1 , . . . , gs (∈ G) be a system of representatives for GP \G.

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Then, V = si=1 (K#D1 )gi u. Fix 1 ≤ i ≤ s, and let w = (w1 , . . . , wn ) ∈ An1 be the natural image of gi u. It sufﬁces to prove that W := (K#D1 )w has ﬁnite K-dimension. By renumbering, we suppose that w1 , . . . , wr (r ≤ n) are a k-basis of the k-subspace of A1 spanned by w1 , . . . , wn . Set w = (w1 , . . . , wr ). Then, w = w T for some rank r matrix T with entries in k. It sufﬁces to prove that W := (K#D1 )w has ﬁnite → W via the right multiplication by T , but this follows by an K-dimension, since W − obvious modiﬁcation of the proof of 8.8, (i) ⇒ (ii), since w1 , . . . , wr form a k-basis ¯ of an H-subcomodule of An1 . (2) We can reprove 3.15 (1) in the present context. The result, together with Part 1 above and 12.6 (3), proves Part 2; see the proof of 3.17. We can now reprove Theorem 8.11 in the present context; see [11, Theorem 4.10]. Theorem 13.4 (Unique existence). Suppose that K is an AS D-module algebra such that the ﬁeld KD is algebraically closed. Let V be a K#D-module of ﬁnite K-rank. There exists a minimal splitting algebra L for V with LD = KD so that L/K is necessarily a ﬁnitely generated PV extension of AS D-module algebras. Such an L is unique up to isomorphism of D-module algebras over K. Proof. Modify the proof of 4.3 by using the following lemma, which generalizes 4.2. Lemma 13.5 (cf. [21, Appendix]). Let A/K be an extension of D-module algebra such that A is ﬁnitely generated as a K-algebra. Suppose that K is AS, and A is simple; then by 11.5 (3), A GP (A/P), where P ∈ (A). Let L = Q(A); by 10.4, L is uniquely a D-module algebra, which is AS. Then the ﬁeld extension LD /KD is algebraic. Proof. As in the ﬁrst part of the proof of 4.2, one sees LD = AD . The remaining parts are also valid, since by 11.1 and 11.5 (3), we can suppose that K is a ﬁeld, and A is a domain, by replacing K, A with K/P ∩ K, A/P, where P ∈ (A). We can rewrite Section 5 in the present context. We only point out some aspects; see [13] for details. Let L/K be an extension of AS D-module algebras. Recall the remark on the Amitsur cohomology vanishing just after Lemma 1.14. One then sees that the lemma holds for L/K, since K is the product of ﬁelds, and L is K-free. Using the lemma we can restate Example 1.15. To proceed, understand Ks to denote (K1s ) with the notation of 12.8, where K1s denotes the separable algebraic closure of K1 in the ﬁeld extension L1 /K1 . References [1] I. Kaplansky, “An introduction to differential algebra”, Second ed., Hermann, Paris, 1976. [2] E.R. Kolchin, “Differential algebra and algebraic groups”, Pure Appl. Math., vol. 54, Academic Press, New York-London, 1973.

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[3] A.R. Magid, “Lectures on differential Galois theory”, Univ. Lec. Series vol. 7, AMS, Providence, 1994. [4] M. van der Put, M.F. Singer, “Galois theory of linear differential equations”, Grundlehren Math. Wiss., vol. 328, Springer, Berlin, 2003. [5] E. R. Kolchin, Picard-Vessiot theory of partial differential ﬁelds, Proc. Amer. Math. Soc. 3 (1952) 596–603. [6] A. Buium, “Differential function ﬁelds and moduli of algebraic varieties”, Lecture Notes in Math., vol. 1226, Springer, Berlin, 1986. [7] K. Okugawa, Basic properties of differential ﬁelds of an arbitrary characteristic and the PicardVessiot theory, J. Math. Kyoto Univ. 2–3 (1963) 295–322. [8] C.H. Franke, Picard-Vessiot theory of linear homogeneous difference equations, Trans. Amer. Math. Soc. 108 (1963) 491–515. [9] M. van der Put, M.F. Singer, “Galois theory of difference equations”, Lecture Notes in Math. vol. 1666, Springer, Berlin, 1997. [10] M. Takeuchi, A Hopf algebraic approach to the Picard-Vessiot theory, J. Algebra 122 (1989) 481–509. [11] K. Amano, A. Masuoka, Picard-Vessiot extensions of artinian simple module algebras, J. Algebra 285 (2005) 743–767. [12] Y. André, Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. Éc. Norm. Super. 34 (3) (2001) 685–739; arXiv:math.GM/0203274. [13] K. Amano, Liouville extensions of artinian simple module algebras, Commun. Algebra 34 (2006) 1811–1823. [14] S. Montgomery, “Hopf algebras and their actions on rings”, CBMS Reg. Conf. Series in Math., vol. 82, AMS, 1993. [15] P. Deligne, Catégories Tannakiennes, in P. Cartier, et al., (eds.), “Grothendieck Festschrift”, vol. 2, Progress in Math. 87, Birkhäuser, Boston, 1990, pp. 111–195. [16] M. Sweedler, “Hopf algebras”, Benjamin, New York, 1969. [17] H.F. Kreimer, M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981) 675–692. [18] M. Takeuchi, A correspondance between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972) 251–270. [19] W.C. Waterhouse, “Introduction to afﬁne group schemes”, Grad. Texts in Math., vol. 66, Springer, New York-Berlin, 1979. [20] M. Sweedler, The predual theorem to the Jacobson-Bourbaki theorem, Trans. Amer. Math. Soc. 213 (1975) 391–406. [21] A.H.M. Levelt, Differential Galois theory and tensor products, Indag. Math. (N.S.), 1 (4) (1990) 439–450. [22] Y. Doi, M. Takeuchi, Hopf-Galois extensions of algebras, the Miyashita-Ulbrich action, and Azumaya algebras, J. Algebra 121 (1989) 488–516. [23] R.G. Heyneman, M. Sweedler, Afﬁne Hopf algebras, II, J. Algebra 16 (1970) 271–297. [24] R.M. Cohn, “Difference algebra”, Interscience Tracts Pure Appl. Math., vol. 17, Interscience Publ., New York-London-Sydney, 1965. [25] Y. Doi, M. Takeuchi, Cleft comodule algebras for a bialgebra, Commun. Algebra 14 (1986) 801–818.

Hopf Algebroids Gabriella Böhm Research Institute for Particle and Nuclear Physics, Budapest, H-1525 Budapest 114, P.O.B.49 Hungary E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. R-rings and R-corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. R-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. R-corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bialgebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Right and left bialgebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Construction of new bialgebroids from known ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. The monoidal category of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. The monoidal category of comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Algebra extensions by bialgebroids. Galois extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Hopf algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Examples and constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Basic properties of Hopf algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Comodules of Hopf algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Integral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Galois theory of Hopf algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Alternative notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction If it should be formulated in one sentence what a Hopf algebroid is, it should be described as a generalization of a Hopf algebra to a noncommutative base algebra. More precisely, the best known examples of Hopf algebroids are Hopf algebras. Clearly, the notion of a Hopf algebroid turns out to be a successful generalization only if a convincing amount of results about Hopf algebras extend to Hopf algebroids. But one expects more: working with Hopf algebroids should be considered to be useful if in this way one could solve problems that could not be solved in terms of Hopf algebras. Hopf algebroids provide us with results of two types: one that extends known results about Hopf algebras and the other that is conceptually new. Hopf algebras have been intensively studied and successfully applied in various ﬁelds of mathematics and even physics, for more than 50 years. Without aiming at a complete list, let us mention a few applications. Hopf algebras were used to construct invariants in topology and knot theory. In connection with solutions of the quantum Yang–Baxter equation, quantum groups, that is, certain Hopf algebras, play a central role. In (low-dimensional) quantum ﬁeld theory, Hopf algebras are capable of describing internal symmetry of some models. In noncommutative differential geometry (faithfully ﬂat), Galois extensions by a Hopf algebra are interpreted as noncommutative principal bundles. Although the theory of Hopf algebras was (is!) extremely successful, in the 1990s, there arose more and more motivations for a generalization. Originally, the term Hopf algebroid was used for cogroupoid objects in the category of commutative algebras. These are examples of Hopf algebroids in the current chapter with commutative underlying algebra structure. They found an application, for example, in algebraic topology [1]. As a tool for a study of the geometry of principal ﬁber bundles with groupoid symmetry, recently, more general, noncommutative Hopf algebroids have been used but still over commutative base algebras [2]. For some applications, this is still not the necessary level of generality. In Poisson geometry, solutions of the dynamical Yang–Baxter equation correspond to dynamical quantum groups, which are not Hopf algebras [3–8]. In topology, invariants obtained in [9] do not ﬁt the Hopf algebraic framework. In transverse geometry, extensions of Hopf algebras by noncommutative base algebras occurred [10]. In low-dimensional quantum ﬁeld theories, nonintegral values of the statistical (also called quantum) dimensions in some models exclude a Hopf algebra symmetry [11]. Another ﬁeld where important questions could not be answered in the framework of Hopf algebras is noncommutative geometry, that is, Hopf Galois theory. Thinking about classical Galois extensions of ﬁelds by a ﬁnite group, such an extension can be characterized without explicitly mentioning the Galois group. A (unique, up to isomorphism) Galois group is determined by a Galois ﬁeld extension. In the case of Hopf Galois extensions, no such intrinsic characterization, without explicit use of a Hopf algebra, is known. Also, although the Hopf algebra describing the symmetry of a given Hopf Galois extension is known to be nonunique, the relation between the possible choices is not known. These questions have been handled by allowing for noncommutative base algebras [12, 13]. On the other hand, as the study of Hopf algebroids has a quite short

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past, there are many aspects of Hopf algebras that have not yet been investigated as to how they extend to Hopf algebroids. It has to be admitted that almost nothing has been done yet toward a classiﬁcation and structure theory of Hopf algebroids. What does it mean that the base algebra R of a Hopf algebroid is noncommutative? Recall that a bialgebra over a commutative base ring k is a k-module, with compatible algebra and coalgebra structures. By analogy, in a bialgebroid, the coalgebra structure is replaced by a coring over any not necessarily commutative k-algebra R. Also, the algebra structure is replaced by a ring over a noncommutative base algebra. However, in order to formulate the compatibility between the ring and coring structures, the base algebra of the ring has to be not R but R ⊗k Rop . A Hopf algebra is a bialgebra with an additional antipode map. In the Hopf algebroid case, the antipode relates two different bialgebroid structures over the base algebras R and Rop . In these notes, we arrive at the notion of a Hopf algebroid after considering all constituent structures. In Section 2, R-rings and R-corings are introduced. They are seen to generalize algebras and coalgebras. Emphasis is put on their duality. Section 3 is devoted to a study of bialgebroids, generalizing bialgebras. Several equivalent descriptions are given and examples are collected. In particular, constructions of new bialgebroids from known ones are presented. Some of them change the base algebra of a bialgebroid, so they have no counterparts for bialgebras. Although bialgebroid axioms are not manifestly self-dual, duals of ﬁnitely generated and projective bialgebroids are shown to be bialgebroids. Key properties of a bialgebroid are monoidality of the categories of modules and comodules. This is explained in some detail. Section 3 is closed by a most important and most successful application, Galois theory of bialgebroids. Hopf algebroids are the subject of Section 4. After presenting the deﬁnition, listing some examples, and deriving some immediate consequences of the axioms, we discuss the theory of comodules. Since in a Hopf algebroid there are two bialgebroids (hence corings), comodules of the Hopf algebroid comprise comodule structures of both. The relation between the categories of comodules of a Hopf algebroid and comodules of the constituent bialgebroids is investigated. The category of comodules of a Hopf algebroid is proven to be monoidal, what is essential from the point of view of Galois theory. Next, we turn to a study of the theory of integrals. It is a good example of results that are obtained by using some new ideas but that extend analogous results for Hopf algebras in a reassuring way. The structure of Galois extensions by Hopf algebroids is investigated. Useful theorems are presented about situations when surjectivity of a canonical map implies Galois property. They extend known results about Hopf Galois extensions. While there seems to be an accord in the literature that the right generalization of a bialgebra to a noncommutative base is a bialgebroid, there is some discussion about the right generalization of a Hopf algebra. We close these notes by collecting and comparing notions suggested by various authors. To keep the list of references perspicuous, we do not refer in these notes to papers containing classical results about Hopf algebras, which are generalized here. We believe it is more useful to give here a detailed bibliography of those papers that deal with structures over a noncommutative base. A very good and detailed bibliography of the literature of Hopf algebras can be found in chapter “Hopf Algebras” of Handbook of Algebra [14].

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Notations and conventions. Throughout k is an associative and commutative unital ring. All algebras are associative and unital k-algebras. A k-algebra is denoted by A, and the underlying k-module is denoted by A. On elements of A, multiplication is denoted by juxtaposition. The unit element is denoted by 1A . For an algebra A, with multiplication (a, a ) → aa , Aop denotes the opposite of A. As a k-module it is equal to A and multiplication is (a, a ) → a a. The category of right (resp. left) modules of an algebra A is denoted by MA (resp. A M). Hom sets are denoted by HomA (−, −) (resp. A Hom(−, −)). The category of A-bimodules is denoted by A MA , and its hom sets by A Hom A (−, −). Often, we identify left A-modules with right Aop -modules, but in every such case, this is explicitly said. Action on a (say right) module M of a k-algebra A, if evaluated on elements m ∈ M and a ∈ A, is denoted by M : m ⊗ a → m · a. For coproducts in a coalgebra and, more generally, in a coring C, Sweedler’s index notation is used. That is, for an element c ∈ C, we write c → c(1) ⊗c(2) (or sometimes c → c(1) ⊗c(2) ) for the coproduct, where implicit summation is understood. Similarly, for a (say right) coaction on a comodule M of a coring, evaluated on an element m ∈ M, the notation M : m → m[0] ⊗ m[1] (or m → m[0] ⊗ m[1] ) is used, where implicit summation is understood. The category of right (resp. left) comodules of a coring C is denoted by MC (resp. C M). Hom sets are denoted by HomC (−, −) (resp. C Hom(−, −)). In any category A, the identity morphism at an object A is denoted by the same symbol A. Hom sets in A are denoted by HomA (−, −). In a monoidal category (M, ⊗, U), we allow for nontrivial coherence isomorphisms (− ⊗ −) ⊗ − ∼ = − ⊗ (− ⊗ −) and − ⊗ U ∼ = − ∼ = U ⊗ − but do not denote them explicitly. (Such monoidal categories are called in the literature, sometimes, lax monoidal.) The opposite of a monoidal category (M, ⊗, U), denoted by (M, ⊗, U)op , means the same category M with opposite monoidal product. A functor F between monoidal categories (M, ⊗, U) and (M , ⊗ , U ) is said to be monoidal if there exist natural transformations F2 : F(−) ⊗ F(−) → F(− ⊗ −) and F0 : U → F(U), satisfying the usual compatibility conditions. F is said to be op-monoidal if there exist compatible natural transformations F2 : F(− ⊗ −) → F(−) ⊗ F(−) and F0 : F(U) → U . A monoidal functor (F, F2 , F0 ) is strong monoidal if F2 and F0 are isomorphisms, and it is strict monoidal if F2 and F0 are identity morphisms. 2. R-rings and R-corings A monoid in a monoidal category (M, ⊗, U) is a triple (A, μ, η). Here, A is an object and μ : A ⊗ A → A and η : U → A are morphisms in M, satisfying associativity and unitality conditions μ◦(μ⊗A) = μ◦(A⊗μ)

and

μ◦(η⊗A) = A = μ◦(A⊗η). (2.1)

The morphism μ is called a multiplication (or product) and η is called a unit. An algebra over a commutative ring k can be described as a monoid in the monoidal category (Mk , ⊗k , k) of k-modules.

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A right module of a monoid (A, μ, η) is a pair (V, ν), where V is an object and ν : V ⊗ A → V is a morphism in M such that ν ◦ (V ⊗ μ) = ν ◦ (ν ⊗ A)

and

ν ◦ (V ⊗ η) = V.

Left modules are deﬁned symmetrically. Reversing all arrows in the deﬁnition of a monoid, we arrive at the dual notion of a comonoid. A comonoid in a monoidal category (M, ⊗, U) is a triple (C, , ). Here, C is an object and : C → C ⊗ C and : C → U are morphisms in M, satisfying coassociativity and counitality conditions (⊗C)◦ = (C⊗)◦

and

(⊗C)◦ = C = (C⊗)◦. (2.2)

The morphism is called a comultiplication (or coproduct) and is called a counit. A coalgebra over a commutative ring k can be described as a comonoid in the monoidal category (Mk , ⊗k , k) of k-modules. Dualizing the deﬁnition of a module of a monoid, one arrives at the notion of a comodule of a comonoid. Many aspects of the theory of algebras and their modules, or coalgebras and their comodules, can be extended to monoids or comonoids in general monoidal categories. Here, we are interested in monoids and comonoids in a monoidal category (R MR , ⊗R , R) of bimodules over a k-algebra R. These monoids and comonoids are called R-rings and R-corings, respectively. 2.1. R-rings Generalizing algebras over commutative rings, we study monoids in bimodule categories. Deﬁnition 2.1. For an algebra R over a commutative ring k, an R-ring is a triple (A, μ, η). Here, A is an R-bimodule and μ : A ⊗R A → A and η : R → A are R-bimodule maps, satisfying the associativity and unit conditions (2.1). A morphism of R-rings f : (A, μ, η) → (A , μ , η ) is an R-bimodule map f : A → A such that f ◦ μ = μ ◦ (f ⊗R f) and f ◦ η = η . For an R-ring (A, μ, η), the opposite means the Rop -ring (Aop , μop , η). Here, Aop is the same k-module A. It is understood to be a left (resp. right) Rop -module via the right (resp. left) R-action. Multiplication is μop (a ⊗Rop a ) := μ(a ⊗R a) and unit is η. A most handy characterization of R-rings comes from the following observation. Lemma 2.2. There is a bijective correspondence between R-rings (A, μ, η) and k-algebra homomorphisms η : R → A. Indeed, starting with an R-ring (A, μ, η), a multiplication map A ⊗k A → A is obtained by composing the canonical epimorphism A ⊗k A → A ⊗R A with μ. Conversely, starting with an algebra map η : R → A, an R-bilinear associative

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multiplication A ⊗R A → A is obtained by using the universality of the coequalizer A ⊗k A → A ⊗R A. An R-ring (A, μ, η) determines monads on the categories of right and left R-modules (i.e. monoids in the monoidal categories of endofunctors on MR and R M). They are given by − ⊗R A : MR → MR and A ⊗R − : R M → R M, respectively. Deﬁnition 2.3. A right module for an R-ring (A, μ, η) is an algebra for the monad − ⊗R A on the category MR . A right module morphism is a morphism of algebras for the monad − ⊗R A. A left module for an R-ring (A, μ, η) is an algebra for the monad A ⊗R − on the category R M. A left module morphism is a morphism of algebras for the monad A ⊗R −. Left modules of an R-ring are canonically identiﬁed with right modules for the opposite Rop -ring. Analogously to Lemma 2.2, modules for R-rings can be characterized as follows. Lemma 2.4. A k-module M is a (left or right) module of an R-ring (A, μ, η) if and only if it is a (left or right) module of the corresponding k-algebra A in Lemma 2.2. Furthermore, a k-module map f : M → M is a morphism of (left or right) modules of an R-ring (A, μ, η) if and only if it is a morphism of (left or right) modules of the corresponding k-algebra A in Lemma 2.2. The situation when the (left or right) regular R-module extends to a (left or right) module of an R-ring (A, μ, η) is of particular interest. Lemma 2.5. The right regular module of a k-algebra R extends to a right module of an R-ring (A, μ, η) if and only if there exists a k-module map χ : A → R, obeying the following properties. (i) χ(aη(r)) = χ(a)r, for a ∈ A and r ∈ R (right R-linearity), (ii) χ(aa ) = χ((η ◦ χ)(a)a ), for a, a ∈ A (associativity), (iii) χ(1A ) = 1R (unitality). A map χ obeying these properties is called a right character on the R-ring (A, μ, η). In terms of a right character χ, a right A-action on R is given by r · a := χ(η(r)a). Conversely, in terms of a right A-action on R, a right character is constructed as χ(a) := 1R · a. Symmetrically, one can deﬁne a left character on an R-ring (A, μ, η) via the requirement that the left regular R-module extends to a left module for (A, μ, η). Deﬁnition 2.6. Let (A, μ, η) be an R-ring possessing a right character χ : A → R. The invariants of a right module (M, M ) with respect to χ are the elements of the

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k-submodule Mχ := { m ∈ M | M (m ⊗ a) = M (m ⊗ (η◦χ)(a)), R

R

∀a ∈ A } ∼ = HomA (R, M),

where the isomorphism Mχ → HomA (R, M) is given by m → (r → m · η(r)). In particular, the invariants of R are the elements of the subalgebra B := Rχ = { b ∈ R | χ(η(b)a) = bχ(a),

∀a ∈ A }.

Associated to a character χ, there is a canonical map A → B End(R), a → r → χ(η(r)a) .

(2.3)

The R-ring (A, μ, η) is said to be a Galois R-ring (with respect to the character χ) provided that the canonical map (2.3) is bijective. 2.2. R-corings The theory of R-corings is dual to that of R-rings. A detailed study can be found in the monograph [15]. Deﬁnition 2.7. For an algebra R over a commutative ring k, an R-coring is a triple (C, , ). Here, C is an R-bimodule and : C → C ⊗R C and : C → R are R-bimodule maps, satisfying the coassociativity and counit conditions (2.2). A morphism of R-corings f : (C, , ) → (C , , ) is an R-bimodule map f : C → C such that ◦ f = (f ⊗R f) ◦ and ◦ f = . For an R-coring (C, , ), the co-opposite is the Rop -coring (Ccop , cop , ). Here Ccop is the same k-module C. It is understood to be a left (resp. right) Rop -module via the right (resp. left) R-action. The comultiplication is cop (c) := c(2) ⊗Rop c(1) and the counit is . An R-coring (C, , ) determines comonads on the categories of right and left R-modules (i.e. comonoids in the monoidal categories of endofunctors on MR and R M). They are given by − ⊗R C : MR → MR and C ⊗R − : R M → R M, respectively. Deﬁnition 2.8. A right comodule for an R-coring (C, , ) is a coalgebra for the comonad − ⊗R C on the category MR . That is, a pair (M, M ), where M is a right R-module and M : M → M ⊗R C is a right R-module map, satisfying the coassociativity and counit conditions (M ⊗ C) ◦ M = (M ⊗ ) ◦ M R

R

(M ⊗ ) ◦ M = M.

and

R

(2.4)

A right comodule morphism f : (M, M ) → (M , M ) is a morphism of coalgebras for the comonad − ⊗R C. That is, a right R-module map f : M → M , satisfying M ◦ f = (f ⊗R C) ◦ M .

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Symmetrically, a left comodule is a coalgebra for the comonad C ⊗R − on the category R M.Aleft comodule morphism is a morphism of coalgebras for the comonad C ⊗R −. Left comodules of an R-coring are canonically identiﬁed with right comodules for the co-opposite Rop -coring. The situation when the (left or right) regular R-module extends to a (left or right) comodule of an R-coring (C, , ) is of particular interest (see [16, Lemma 5.1]). Lemma 2.9. The (left or right) regular R-module extends to a (left or right) comodule of an R-coring (C, , ) if and only if there exists an element g ∈ C, obeying the following properties. (i) (g) = g ⊗R g. (ii) (g) = 1R . An element g obeying these properties is called a grouplike element in the R-coring (C, , ). Having a grouplike element g in an R-coring (C, , ), a right coaction on R is constructed as a map R → C, r → g · r. Conversely, a right coaction R : R → C determines a grouplike element R (1R ). Deﬁnition 2.10. Let (C, , ) be an R-coring possessing a grouplike element g ∈ C. The coinvariants of a right comodule (M, M ) with respect to g are the elements of the k-submodule M g := { m ∈ M | M (m) = m ⊗ g } ∼ = HomC (R, M), R

HomC (R, M)

Mg

where the isomorphism → is given by m → (r → m · r). In particular, the coinvariants of R are the elements of the subalgebra B := Rg = { b ∈ R | b · g = g · b }. Associated to a grouplike element g, there is a canonical map R ⊗ R → C, B

r ⊗ r → r · g · r . B

(2.5)

The R-coring (C, , ) is said to be a Galois R-coring (with respect to the grouplike element g) provided that the canonical map (2.5) is bijective. Let (C, , ) be an R-coring possessing a grouplike element g. Put B := Rg . For any right C-comodule M, M g is a right B-module. Furthermore, any right C-comodule map M → M restricts to a right B-module map M g → M g . There is an adjoint pair of functors − ⊗ R : MB → MC B

and

(−)g : MC → MB .

(2.6)

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If (C, , ) is a Galois coring (with respect to g), then MC is equivalent to the category of descent data for the extension B ⊆ R. Hence, the situation, when the functors (2.6) establish an equivalence, is interesting from the descent theory point of view. 2.3. Duality Beyond the formal duality between algebras and coalgebras, it is well known that the k-dual of a coalgebra over a commutative ring k possesses a canonical algebra structure. The converse is true whenever a k-algebra is ﬁnitely generated and projective as a k-module. In what follows, we recall analogs of these facts for rings and corings over an arbitrary algebra R. Proposition 2.11. Let R be an algebra over a commutative ring k. (1) For an R-coring (C, , ), the left dual ∗ C := R Hom(C, R) possesses a · canonical R-ring structure. Multiplication is given by (φψ)(c) := ψ c (1) φ(c(2) ) , for φ, ψ ∈ ∗ C and c ∈ C. Unit map is R → ∗ C, r → (−)r. (2) For an R-ring (A, μ, η), which is a ﬁnitely generated and projective right Rmodule, the right dual A∗ := HomR (A, R) possesses a canonical R-coring ∗ structure. In terms of a dual basis ({ai ∈ A}, {αi ∈ A }), comultiplication is given by ξ → i ξ(ai −) ⊗R αi , which is independent of the choice of a dual basis. Counit is A∗ → R, ξ → ξ(1A ). (3) For an R-coring (C, , ), which is a ﬁnitely generated and projective left R-module, the second dual (∗ C)∗ is isomorphic to C as an R-coring. (4) For an R-ring (A, μ, η), which is a ﬁnitely generated and projective right R-module, the second dual ∗ (A∗ ) is isomorphic to A as an R-ring. Applying Proposition 2.11 to the co-opposite coring and the opposite ring, analogous correspondences are found between right duals of corings and left duals of rings. Proposition 2.12. Let C be a coring over an algebra R. (1) Any right C-comodule (M, M ) possesses a right module structure for the R-ring ∗ C, m · φ := m[0] · φ(m[1] ),

for m ∈ M, φ ∈ ∗ C.

(2.7)

Any right C-comodule map becomes a ∗ C-module map. That is, there is a faithful functor MC → M ∗C . (2) The functor MC → M ∗C is an equivalence if and only if C is a ﬁnitely generated and projective left R-module. There is a duality between Galois rings and Galois corings too.

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Proposition 2.13. Let C be an R-coring, which is a ﬁnitely generated and projective left R-module. For an element g ∈ C, introduce the map χg : ∗ C → R, φ → φ(g). The following statements hold. (1) The element g ∈ C is grouplike if and only if χg is a right character on the R-ring ∗ C. (2) An element b ∈ R is a coinvariant of the right C-comodule R (with coaction induced by a grouplike element g) if and only if b is an invariant of the right ∗ C-module R (with respect to the right character χ ). g (3) The R-coring C is a Galois coring (with respect to a grouplike element g) if and only if the R-ring ∗ C is a Galois ring (with respect to the right character χg ). 3. Bialgebroids In Section 2, we generalized algebras and coalgebras over commutative rings to monoids and comonoids in bimodule categories. We could easily do so, the category of bimodules over any k-algebra R is a monoidal category, just as the category of k-modules. If we try to generalize bialgebras to a noncommutative base algebra R, however, we encounter difﬁculties. Recall that a k-bialgebra consists of an algebra (B, μ, η), and a coalgebra (B, , ) deﬁned on the same k-module B. They are subject to compatibility conditions. Unit and multiplication must be coalgebra maps or equivalently counit, and comultiplication must be algebra maps. This means in particular that, for any elements b and b in B, multiplication and comultiplication satisfy the condition ⊗ b(2) b(2) . (bb )(1) ⊗ (bb )(2) = b(1) b(1) k

(3.1)

k

Note that (3.1) is formulated in terms of the symmetry tw in Mk . For any k-modules M and N, the twist map twM,N : M ⊗k N → N ⊗k M maps m ⊗k n to n ⊗k m. Written in terms of morphisms (3.1) is equivalent to ◦ μ = (μ ⊗ μ) ◦ (B ⊗ twB,B ⊗ B) ◦ ( ⊗ ). k

k

k

k

In the literature, one can ﬁnd generalizations when tw is replaced by a braiding [17–19]. (For an approach when tw is replaced by a mixed distributive law (see [20]).) However, general bimodule categories are neither symmetric nor braided. There is no natural way to formulate an analog of (3.1) in a bimodule category. Indeed, more sophisticated ideas are needed. The notion that is known today as a (left) bialgebroid was introduced (independently) by several authors. The ﬁrst deﬁnition is due to Takeuchi, who used the name ×R -bialgebra in [21]. Some twenty years later, with motivation coming from Poisson geometry, in [6], Lu proposed an equivalent deﬁnition. The term bialgebroid is due to her. A third equivalent set of axioms was invented by Xu in [8]. The equivalence of the listed deﬁnitions is far from obvious. It was proven by Brzezin´ ski and Militaru

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in [22]. Symmetrical notions of left and right bialgebroids were formulated and studied by Kadison and Szlachányi in [23]. The deﬁnition presented here is a slightly reformulated version of the one in [6] or [23]. 3.1. Right and left bialgebroids In contrast to the deﬁnition of a bialgebra, in this section, a bialgebroid is not described as a compatible monoid and comonoid in some monoidal category (of bimodules). The ring and coring structures of a bialgebroid are deﬁned over different base algebras; they are a monoid and a comonoid in different monoidal categories. Recall from Lemma 2.2 that an R ⊗k Rop -ring A (for some algebra, R over a commutative ring k) is described by a k-algebra map η : R ⊗k Rop → A. Equivalently, instead of η, we can consider its restrictions s := η(− ⊗ 1R ) : R → A

and

k

t := η(1R ⊗ −) : Rop → A, (3.2) k

which are k-algebra maps with commuting ranges in A. The maps s and t in (3.2) are called the source and target maps of an R ⊗k Rop -ring A, respectively. In what follows, an R ⊗k Rop -ring will be given by a triple (A, s, t), where A is a k-algebra (with underlying k-module A) and s and t are algebra maps with commuting ranges as in (3.2). Deﬁnition 3.1. Let R be an algebra over a commutative ring k. Aright R-bialgebroid B consists of an R ⊗k Rop -ring (B, s, t) and an R-coring (B, , ) on the same k-module B. They are subject to the following compatibility axioms. (i) The bimodule structure in the R-coring (B, , ) is related to the R ⊗k Rop ring (B, s, t) via r · b · r := bs(r )t(r),

for r, r ∈ R, b ∈ B.

(3.3)

(ii) Considering B as an R-bimodule as in (3.3), the coproduct corestricts to a k-algebra map from B to B×R B :=

i

bi ⊗ bi R

|

i

s(r)bi ⊗ bi R

=

i

bi ⊗ t(r)bi , R

∀r ∈ R

,

(3.4) where B ×R B is an algebra via factorwise multiplication. (iii) The counit is a right character on the R-ring (B, s). Remark 3.2. The bialgebroid axioms in Deﬁnition 3.1 have some immediate consequences. (1) Note that the k-submodule B ×R B of B ⊗R B is deﬁned in such a way that factorwise multiplication is well deﬁned on it. B ×R B is called the Takeuchi

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product. In fact, it has more structure than that of a k-algebra: it is an R ⊗k Rop ring with unit map R ⊗k Rop → B ×R B, r ⊗k r → t(r ) ⊗R s(r). The (corestriction of the) coproduct is easily checked to be an R ⊗k Rop -bimodule map B → B ×R B. (2) Axiom (iii) is equivalent to the requirement that the counit is a right character on the Rop -ring (B, t). (3) Yet another equivalent formulation of axiom (iii) is the following. The map θ : B → End k (R)op , b → r → (s(r)b) is a k-algebra map, where End k (R)op is an algebra via opposite composition of endomorphisms. The map θ is called an anchor map in [8]. Recall that in a bialgebra over a commutative ring, replacing the algebra with the opposite one or replacing the coalgebra with the co-opposite one, one arrives at bialgebras again. Analogously, the co-opposite of a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1, is the following right Rop -bialgebroid Bcop . The Rop ⊗k R-ring structure is (B, t, s), and the Rop -coring structure is (Bcop , cop , ). However, the R ⊗k Rop -ring (Bop , t, s) and the R-coring (B, , ) do not satisfy the same axioms in Deﬁnition 3.1. Instead, they are subject to a left- right symmetrical version of Deﬁnition 3.1. Deﬁnition 3.3. Let R be an algebra over a commutative ring k. A left R-bialgebroid B consists of an R ⊗k Rop -ring (B, s, t) and an R-coring (B, , ) on the same k-module B. They are subject to the following compatibility axioms. (i) The bimodule structure in the R-coring (B, , ) is related to the R ⊗k Rop ring (B, s, t) via r · b · r := s(r)t(r )b,

for r, r ∈ R, b ∈ B.

(3.5)

(ii) Considering B as an R-bimodule as in (3.5), the coproduct corestricts to a k-algebra map from B to B R×B := bi ⊗ b i | bi t(r) ⊗ bi = bi ⊗ bi s(r), ∀r ∈ R , i

R

i

R

i

R

(3.6) where B R× B is an algebra via factorwise multiplication. (iii) The counit is a left character on the R-ring (B, s). Since in this note left and right bialgebroids are considered simultaneously, we use two versions of Sweedler’s index notation. In a left bialgebroid, we use lower indices to denote components of the coproduct, that is, we write b → b(1) ⊗R b(2) . In a right bialgebroid, we use upper indices to denote components of the coproduct, that is, we write b → b(1) ⊗R b(2) . In both cases, implicit summation is understood. Recall that coalgebras over a commutative ring k form a monoidal category with respect to the k-module tensor product. Bialgebras over k can be described as monoids in the monoidal category of k-coalgebras. In [21], Takeuchi deﬁned bialgebroids

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(×R -bialgebras in his terminology) as monoids in a monoidal category of certain corings too. By [24, Deﬁnition 3.5], for two k-algebras R and S, an S|R-coring is an S ⊗k R-bimodule C together with an R-coring structure (C, , ) such that the following identities hold. ((s ⊗ 1R ) · c · (s ⊗ 1R )) = c(1) · (s ⊗ 1R ) ⊗ (s ⊗ 1R ) · c(2) k

k

k

k

R

(s ⊗ 1R ) · c(1) ⊗ c(2) = c(1) ⊗ c(2) · (s ⊗ 1R ), k

R

k

R

and

for s, s ∈ S, c ∈ C.

Morphisms of S|R-corings are morphisms of R-corings, which are in addition S-bimodule maps. In particular, it can be shown by using the same methods as in [22] that the notion of an R|R-coring is equivalent to a ×R -coalgebra in [21, Deﬁnition 4.1]. Part (1) of the following Theorem 3.4 is, thus, a reformulation of [21, Proposition 4.7]. Part (2) states an equivalence of a right Rbialgebroid as in Deﬁnition 3.1 and a symmetrical version of a ×R -bialgebra as in [21, Deﬁnition 4.5]. Theorem 3.4. For an algebra R over a commutative ring k, the following statements hold. (1) R|R-corings form a monoidal category. The monoidal product of two objects (C, , ) and (C , , ) is the R ⊗k Rop -module tensor product C C := C ⊗ C /{ (1R ⊗ r)·c·(1R ⊗ r ) ⊗ c −c ⊗ (r ⊗ 1R )·c ·(r ⊗ 1R ) | r, r ∈ R }. k

C

C

k

k

k

k

k

k

is an R ⊗k R-bimodule, via the actions

(r1 ⊗ r2 )·(c c )·(r1 ⊗ r2 ) := (r1 ⊗ 1R )·c·(r1 ⊗ 1R ) (1R ⊗ r2 )·c ·(1R ⊗ r2 ). k

k

k

The coproduct and counit in C

C

k

k

are

c c → (c(1) c(1) ) ⊗ (c(2) c(2) ) R

k

and

c c → ((c) ⊗ 1R )·c . k

The monoidal unit is R ⊗k R, with R|R-coring structure described in Section 3.2.3 below. The monoidal product of morphisms αi : (Ci , i , i ) → (Ci , i , i ), for i = 1, 2, is given by (α1 α2 )(c1 c2 ) := α1 (c1 ) α2 (c2 ). (2) Monoids in the monoidal category of R|R-corings are the same as right R-bialgebroids. (3) Monoidal morphisms in the monoidal category of R|R-corings are the same as maps of R ⊗k Rop -rings and of R-corings. Considering a right R-bialgebroid as an R|R-coring, the ·|R -bimodule structure is given by right multiplications by the source and target maps. The R|· -bimodule structure is given by left multiplications by the source and target maps. Theorem 3.4 was extended by Szlachányi in [25]. He has shown that S|R-corings form a bicategory. Bialgebroids can be described as monads in this bicategory. This

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makes it possible to deﬁne (base changing) morphisms of bialgebroids as bimodules for the corresponding monads. The constructions described in Section 3.4.1 and also in Deﬁnition 3.15 provide examples of bialgebroid morphisms in this sense. For more details we refer to [25]. An S|R-coring C can be looked at as an S ⊗k Sop -R ⊗k Rop bimodule in a canonical way. Identifying S-bimodules with right S ⊗k Sop -modules and R-bimodules with right R ⊗k Rop -modules, there is an op-monoidal left adjoint functor − ⊗S⊗k Sop C : S MS → R MR . As a matter of fact, this correspondence establishes a bifunctor from the bicategory of S|R-corings to the 2-category of op-monoidal left adjoint functors between bimodule categories. For any two 0-cells (i.e. algebras S and R), it gives an equivalence of the vertical subcategories. So, in addition to a characterization of bialgebroids as monads in the bicategory of S|R-corings, they can be described as monads in the 2-category of op-monoidal left adjoint functors between bimodule categories. Monads in the 2-category of op-monoidal functors (i.e. op-monoidal monads such that multiplication and unit natural transformations of the monad are compatible with the op-monoidal structure) were termed Hopf monads in [26] and bimonads in [27]. Using the latter terminology, the following characterization of bialgebroids in [27, Theorem 4.5] is obtained. Theorem 3.5. For an algebra R, any right R-bialgebroid induces a bimonad on R MR , which possesses a right adjoint. Conversely, every bimonad on R MR possessing a right adjoint is naturally equivalent to a bimonad induced by a right R-bialgebroid. Another aspect of the equivalence in Theorem 3.5 is explained in Section 3.5. In the paper [28] by Day and Street, op-monoidal monads were studied in the more general framework of pseudomonoids in monoidal bicategories. Based on Theorem 3.5, a description of bialgebroids as strong monoidal morphisms between pseudomonoids in the monoidal bicategory of bimodules was obtained [28, Proposition 3.3]. 3.2. Examples In order to make the reader more familiar with the notion, in this section, we list some examples of bialgebroids. 3.2.1. Bialgebras Obviously, a bialgebra B over a commutative ring k determines a (left or right) k-bialgebroid in which both the source and target maps are equal to the unit map k → B. Note, however, that there are k-bialgebroids in which the source and target maps are different, or possibly they are equal but their range is not central in the total algebra, hence they are not bialgebras (see Section 4.1.4). 3.2.2. Weak bialgebras A weak bialgebra over a commutative ring k consists of an algebra and a coalgebra structure on the same k-module B, subject to compatibility axioms generalizing the axioms of a bialgebra [29, 30]. Explicitly, the coproduct

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is required to be multiplicative in the sense of (3.1). Unitality of the coproduct and multiplicativity of the counit are weakened to the identities ((1B ) ⊗ 1B )(1B ⊗ (1B )) = ( ⊗ B) ◦ (1B ) k

k

k

= (1B ⊗ (1B ))((1B ) ⊗ 1B ) k

k

(b1(1) )(1(2) b ) = (bb ) = (b1(2) )(1(1) b ),

and for b and b ∈ B,

respectively. Here, 1(1) ⊗k 1(2) denotes (1B ) (which may differ from 1B ⊗k 1B ). The map

R : B → B,

b → 1(1) (b1(2) )

is checked to be idempotent. Its range is a subalgebra R of B. B is an R ⊗k Rop ring, with source map given by the inclusion R → B and target map given by the restriction to R of the map B → B, b → (b1(1) )1(2) . Consider B as an R-bimodule via right multiplication by these source and target maps. The coproduct in an R-coring B is obtained by composing : B → B ⊗k B with the canonical epimorphism B ⊗k B → B ⊗R B. It has a counit R . The R ⊗k Rop -ring and R-coring structures constructed on B in this way constitute a right R-bialgebroid. A left Rop -bialgebroid structure in a weak bialgebra B is constructed symmetrically. Its source map is the inclusion map into B of the range subalgebra of the idempotent map

L : B → B,

b → (1(1) b)1(2) .

The coproduct is obtained by composing the weak coproduct : B → B ⊗k B with an appropriate canonical epimorphism to an Rop -module tensor product too. As it was observed by Schauenburg in [31] (see also [32] Sections 1.2 and 1.3), the base algebra R of a weak bialgebra B is Frobenius separable. This means the existence of a k-module map R → k(given by the counit ), possessing a dual basis R ⊗k R such that i ei fi = 1R . The dual-basis property means i ei ⊗k fi ∈ e (f r) = r = i i i i (rei )fi , for all r ∈ R. In a weak bialgebra, a dual basis is given by 1(1) ⊗k R (1(2) ) ∈ R ⊗k R. In [23, 31, 32] also the converse is proved: a k-module map : R → k on the base algebra R of a bialgebroid B, with normalized dual basis i ei ⊗k fi ∈ R ⊗k R, determines a weak bialgebra structure on the underlying k-module B. In [23, Proposition 9.3], any separable algebra over a ﬁeld was proved to be Frobenius separable. Consider a small category C with ﬁnitely many objects. For a commutative ring k, the free k-module generated by the morphisms in C carries a weak bialgebra structure. The product of two morphisms is equal to the composite morphism if they are composable, and zero otherwise. The unit element is the (ﬁnite) sum of the identity morphisms for all objects. Extending the product k-linearly, we obtain a k-algebra. The coproduct is diagonal on all morphisms, that is, (f ) = f ⊗k f . The counit maps every morphism to 1k . Extending the coproduct and the counit k-linearly, we obtain a k-coalgebra. The algebra and coalgebra structures constructed in this way constitute a weak bialgebra.

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3.2.3. The bialgebroid R ⊗ Rop For any algebra R over a commutative ring k, a simplest possible right R-bialgebroid is constructed on the algebra R ⊗k Rop . Source and target maps are given by the inclusions R → R ⊗ Rop , k

r → r ⊗ 1R

and

k

Rop → R ⊗ Rop , k

r → 1R ⊗ r, k

respectively. Coproduct is R ⊗ Rop → (R ⊗ Rop ) ⊗ (R ⊗ Rop ), k

k

R

r

k

r ⊗ r → (1R ⊗ r ) ⊗ (r ⊗ 1R ). k

k

R

k

r r.

→ R, r ⊗k → The corresponding R|R-coring occurred Counit is R ⊗k in Theorem 3.4 (1). The opposite co-opposite of the above construction yields a left Rop -bialgebroid structure on Rop ⊗k R ∼ = R ⊗k Rop . Rop

3.3. Duality In Section 2.3, the duality between R-rings and R-corings has been studied. Now, we shall see how it leads to a duality of bialgebroids. Recall that the axioms of a bialgebra over a commutative ring k are self-dual. That is, the diagrams (in the category Mk of k-modules), expressing the bialgebra axioms, remain unchanged if all arrows are reversed. As a consequence, if a bialgebra B is ﬁnitely generated and projective as a k-module (hence possesses a dual in Mk ), then the dual has a bialgebra structure too, which is the transpose of the bialgebra structure of B. In contrast to bialgebras, the axioms of a bialgebroid are not self-dual in the same sense. Although it follows by the considerations in Section 2.3 that the R-dual of a ﬁnitely generated and projective bialgebroid possesses an R-ring and an R-coring structure, it is not obvious that these dual structures constitute a bialgebroid. The fact that they do indeed was shown ﬁrst by Schauenburg in [33]. A detailed study can be found also in the paper [23] by Kadison and Szlachányi. Our presentation here is closer to [23, Proposition 2.5]. Proposition 3.6. Let R be an algebra. Consider a left bialgebroid B over R, which is a ﬁnitely generated and projective left R-module (via left multiplication by the source map). Then the left R-dual ∗ B := R Hom(B, R) possesses a (canonical) right R-bialgebroid structure. Applying part (1) of Proposition 2.11 to the R-coring (B, , ) underlying B, we ﬁnd the existence of an R-ring structure on ∗ B. Its unit map is ∗ s : R → ∗ B, r → (−)r. Multiplication is given by (ββ )(b) = β t(β(b(2) ))b(1) , for β, β ∈ B, where we used that the right R-module structure in B is given via the target map t of B. Applying (a symmetrical version of ) part (2) of Proposition 2.11 to the R-ring (B, s) underlying B, we deduce the existence of an R-coring structure in ∗ B. It has a bimodule structure r · β · r = β − s(r) r for r, r ∈ R, β ∈ ∗ B.

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In particular, · r = ∗ s(r), for r ∈ R, as expected. A to-be-target-map is deﬁned as ∗ t(r) := r · , for r ∈ R. The counit in the R-coring ∗ B is ∗ : ∗ B → R, β → β(1 ). B The coproduct is given in terms of a dual basis ({ai ∈ B}, {αi ∈ ∗ B}) as ∗

: ∗ B → ∗ B ⊗ ∗ B, R

β →

i

αi ⊗ β(−ai ). R

The right bialgebroid axioms are veriﬁed by direct computations. One can apply Proposition 3.6 to the co-opposite left bialgebroid, which is a ﬁnitely generated and projective right Rop -module via left multiplication by the target map. In this way, one veriﬁes that the right dual B∗ := HomR (B, R) of a left R-bialgebroid B, which is a ﬁnitely generated and projective right R-module, possesses a right R-bialgebroid structure B∗ := (∗ (Bcop ))cop . Note that (conventionally), multiplication is chosen in such a way that it results in right bialgebroid structures on both duals of a left bialgebroid. Applying the constructions to the opposite bialgebroid, left and right duals of a right bialgebroid, which is a ﬁnitely generated and projective R-module on the appropriate side, are concluded to be left bialgebroids. Our convention is to choose (Bop )∗ := (B ∗ )op and ∗ (B op ) := (∗ B)op . 3.4. Construction of new bialgebroids from known ones In addition to the examples in Section 3.2, further (somewhat implicit) examples of bialgebroids are provided by various constructions starting with given bialgebroids. 3.4.1. Drinfel’d twist A Drinfel’d twist of a bialgebra B over a commutative ring k is a bialgebra with the same algebra structure in B and coproduct deformed (or twisted) by an invertible normalized 2-cocycle in B (the so called Drinfel’d element). In this section, we recall analogous Drinfel’d twists of bialgebroids from [25, Section 6.3]. More general twists, which do not correspond to invertible Drinfel’d elements, are studied in [8]. Such generalized twists will not be considered here. Deﬁnition 3.7. For an algebra R, consider a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1. An (invertible) element J of the Takeuchi product B ×R B is called an (invertible) normalized 2-cocycle in B provided it satisﬁes the following conditions. (i) (t(r) ⊗R s(r ))J = J(t(r) ⊗R s(r )) for r, r ∈ R (bilinearity), (ii) (J ⊗R 1B )( ⊗R B)(J) = (1B ⊗R J)(B ⊗R )(J) (cocycle condition), (iii) ( ⊗R B)(J) = 1B = (B ⊗R )(J) (normalization). Proposition 3.8. Let J be an invertible normalized 2-cocycle in a right R-bialgebroid B. The R ⊗k Rop -ring (B, s, t) in B, the counit of B, and the twisted form J := J(−)J −1 of the coproduct in B constitute a right R-bialgebroid BJ . 3.4.2. Cocycle double twist Dually to the construction in Section 3.4.1, one can leave the coproduct in a bialgebroid B unchanged and twist multiplication by an

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invertible normalized 2-cocycle on B. For an algebra R over a commutative ring k, consider a left R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.3. Recall from Theorem 3.4 that the R ⊗k Rop -module tensor product B ⊗R⊗k Rop B (with respect to the right (resp. left) actions given by right (resp. left) multiplications by s and t) is an R-coring. It has a bimodule structure r · (b ⊗ b ) · r := s(r)t(r )b ⊗ b , ) ⊗ (b coproduct b ⊗ b → (b(1) ⊗ b(1) R (2) ⊗ b(2) ), and counit b ⊗ b → (bb ). Hence, there is a corresponding convolution algebra R HomR (B ⊗R⊗k Rop B, R) with )g(b multiplication (f g)(b ⊗ b ) := f(b(1) ⊗ b(1) (2) ⊗ b(2) ). Deﬁnition 3.9. Let R be an algebra over a commutative ring k and let B be a left R-bialgebroid, with structure maps denoted as in Deﬁnition 3.3. An (invertible) element of the convolution algebra R HomR (B⊗R⊗k Rop B, R) is called an (invertible) normalized 2-cocycle on B provided it satisﬁes the following conditions, for b, b , b ∈ B and r, r ∈ R. (i) σ(s(r)t(r )b, b ) = rσ(b, b )r (bilinearity), (ii) σ(b, s(σ(b (1) , b (1) ))b (2) b (2) ) = σ(s(σ(b(1) , b (1) ))b(2) b (2) , b ) condition), (iii) σ(1B , b) = (b) = σ(b, 1B ) (normalization).

(cocycle

Proposition 3.10. Let σ be an invertible normalized 2-cocycle on a left R-bialgebroid B, with inverse σ . The source and target maps s and t in B, the R-coring (B, , ) ) b b , for ) t σ (b(3) , b(3) in B, and the twisted product b ·σ b := s σ(b(1) , b(1) (2) (2) b, b ∈ B, constitute a left R-bialgebroid B σ . 3.4.3. Duality The constructions in Sections 3.4.1 and 3.4.2 are dual of each other in the following sense. Proposition 3.11. For an algebra R, let B be a left R-bialgebroid, which is a ﬁnitely generated and projective right R-module via left multiplication by the target map t, and consider the right dual right R-bialgebroid B∗ . The following statements hold. (1) An element J = k ξk ⊗R ζk ∈ B∗ ×R B∗ is an invertible normalized 2-cocycle in B ∗ if and only if σJ (b, b ) := ξk bt(ζk (b )) (3.7) σJ : B ⊗op B → R, R⊗R

k

is an invertible normalized 2-cocycle on B. (2) Assume the equivalent properties in part (1). The right bialgebroid (B∗ )J , obtained by twisting the coproduct of B∗ by the cocycle J, is right dual of the left bialgebroid B σJ , obtained by twisting the product in B by the cocycle σJ in (3.7). The inverse of the construction in (3.7) is given by associating to an invertible normalized 2-cocycle σ on B an invertible normalized 2-cocycle in B∗ : Jσ := ∗ i σ(−, ai ) ⊗R αi , where ({ai ∈ B}, {αi ∈ B }) is a dual basis.

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3.4.4. Drinfel’d double For a Hopf algebra B over a commutative ring k such that B is a ﬁnitely generated and projective k-module, the k-module D(B) := B ⊗k B∗ has a bialgebra (in fact Hopf algebra) structure. It is known as the Drinfel’d double of B. The category of D(B)-modules is isomorphic (as a monoidal category) to the category of Yetter–Drinfel’d modules for B and also to the monoidal center of the category of B-modules. These results were extended to certain bialgebroids by Schauenburg in [33]. In this section, let B be a left bialgebroid over a k-algebra R, ﬁnitely generated and projective as a right R-module. Assume in addition that the following map is bijective. ϑ : B ⊗op B → B ⊗ B, R

R

b ⊗op b → b(1) ⊗ b(2) b , R

(3.8)

R

where in the domain of ϑ, the module structures are given by right and left multiplications by the target map, and in the codomain, the module structures are given by left multiplications by the target and source maps. Left bialgebroids, for which the map (3.8) is bijective, were named (left) ×R -Hopf algebras in [33] and they are discussed in more detail in Section 4.6.2. In the following Proposition 3.12, Sweedler’s index notation is used for the coproducts and also the index notation ϑ−1 (b ⊗R 1B ) = b1 ⊗Rop b2 , where implicit summation is understood. Proposition 3.12. Let R be an algebra over a commutative ring k. Let B be a left ×R -Hopf algebra, which is a ﬁnitely generated and projective right R-module via left multiplication by the target map. Denote the structure maps in B as in Deﬁnition 3.3. Consider B as a right R ⊗k Rop -module via right multiplications by the source and target maps s and t in the left bialgebroid B. Consider the right dual B∗ as a left R ⊗k Rop -module via right multiplications by the target and source maps t ∗ and s∗ in the right bialgebroid B ∗ . The tensor product D(B) := B ⊗R⊗Rop B∗ has a left R-bialgebroid structure as follows. Multiplication is given by ) b 1 β β(2) s∗ β(3) (b 2 ) , (b β)(b β ) := bs β(1) (b(1) (2) (2) for b β, b β ∈ D(B). The source and target maps are R → D(B),

r → 1B t ∗ (r) and Rop → D(B),

r → 1B s∗ (r),

respectively. The coproduct is b β → (b(1) β(1) ) ⊗ (b(2) β(2) ), R and the counit is D(B) → R, b β → bs(β(1B )) . D(B) → D(B) ⊗ D(B), R

It is not known if the Drinfel’d double (or the dual) of a ×R -Hopf algebra is a ×R -Hopf algebra as well. 3.4.5. Morita base change In contrast to the previous sections, constructions in this section and in the forthcoming ones change the base algebra of a bialgebroid.

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Let R be an algebra over a commutative ring k and let B be a left R-bialgebroid. be a k-algebra, which Denote the structure maps of B as in Deﬁnition 3.3. Let R P, Q, •, ◦). Denote the is Morita equivalent to R. Fix a strict Morita context (R, R, i inverse image of 1R under the map • by i qi ⊗R p ∈ Q⊗R P and denote the inverse j under the map ◦ by p ⊗ q ∈ P ⊗ image of 1R R R Q. The R-R bimodule P j j op op op -Rop determines a canonical R -R bimodule P. Similarly, there is a canonical R bimodule Q. In [34, Section 5], a left R-bialgebroid structure was constructed on the k-module B := (P ⊗k Q) ⊗ B ⊗ (Q ⊗k P), where an unadorned tensor product is meant to be one over R ⊗k Rop . Multiplication is given by [(p1 ⊗ q1 ) ⊗ b ⊗ (q2 ⊗ p2 )][(p1 ⊗ q1 ) ⊗ b ⊗ (q2 ⊗ p2 )] := (p1 ⊗ q1 ) ⊗ bs(q2 • p1 )t(q1 • p2 )b ⊗ (q2 ⊗ p2 ). are, for The source and target maps in B r∈ R, r →

( r·pj ⊗qj )⊗1B ⊗(qj ⊗pj )

and r →

(pj ⊗qj · r )⊗1B ⊗(qj ⊗pj ),

j,j

j,j

respectively. The coproduct and counit are given on an element (p1 ⊗ q1 ) ⊗ b ⊗ (q2 ⊗ B as p2 ) ∈ (p1 ⊗ q1 ) ⊗ b ⊗ (q2 ⊗ p2 ) → [(p1 ⊗ qi ) ⊗ b(1) ⊗ (q2 ⊗ pj )] ⊗ R

i,j

[(pi ⊗ q1 ) ⊗ b(2) ⊗ (qj ⊗ p2 )] (p1 ⊗ q1 ) ⊗ b ⊗ (q2 ⊗ p2 ) → p1 · bs(q2 • p2 ) ◦ q1 ,

and

respectively. A generalization and a more conceptual background of Morita base change in bialgebroids are presented in Section 3.5. 3.4.6. Connes–Moscovici bialgebroids The following bialgebroid was constructed in [10] in the framework of transverse geometry. Let H be a Hopf algebra over a commutative ring k, with coproduct δ : h → h(1) ⊗k h(2) and counit ε. Let R be a left H-module algebra. Consider the k-module B := R ⊗k H ⊗k R. It can be equipped with an associative multiplication (r1 ⊗ h ⊗ r2 )(r1 ⊗ h ⊗ r2 ) := r1 (h(1) · r1 ) ⊗ h(2) h ⊗ (h(3) · r2 )r2 , k

k

k

k

k

k

with unit 1R ⊗k 1H ⊗k 1R . The algebra B can be made a left R-bialgebroid with source and target maps R → B,

r → r ⊗ 1H ⊗ 1R k

r → 1R ⊗ 1H ⊗ r,

Rop → B,

and

k

k

k

respectively. The coproduct and counit are r ⊗ h ⊗ r → (r ⊗ h(1) ⊗ 1R ) ⊗ (1R ⊗ h(2) ⊗ r ) k

k

k

k

R

k

k

and

r ⊗ h ⊗ r → rε(h)r . k

k

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3.4.7. Scalar extension Let B be a bialgebra over a commutative ring k and let A be an algebra in the category of right–right Yetter–Drinfel’d modules of B. Recall that this means A is a right B-module algebra and a right B-comodule algebra such that the following compatibility condition holds, for a ∈ A and b ∈ B. (a · b(2) )[0] ⊗ b(1) (a · b(2) )[1] = a[0] · b(1) ⊗ a[1] b(2) . k

k

The category of right–right Yetter–Drinfel’d modules is prebraided. Assume that A is (a·a ) = aa holds. Under braided commutative, that is, for a, a ∈ A, the identity a[0] [1] these assumptions, it follows by a symmetrical version of [22, Theorem 4.1] that the smash product algebra has a right A-bialgebroid structure. Recall that the smash product algebra is the k-module A#B := A⊗k B, with multiplication (a#b)(a #b ) := ) ⊗ bb . The source and target maps are a (a · b(1) (2) k

s : A → A#B, a → a[0] #a[1]

and t : Aop → A#B, a → a#1B ,

respectively. The coproduct is : A#B → (A#B) ⊗ (A#B), A

a#b → (a#b(1) ) ⊗ (1A #b(2) ) A

and the counit is given in terms of the counit ε in B as A#B → A, a#b → aε(b). The name scalar extension comes from the feature that the base algebra k of B (the subalgebra of “scalars”) becomes replaced by the base algebra A of A#B. A solution R of the quantum Yang–Baxter equation on a ﬁnite-dimensional vector space determines a bialgebra B(R) via the so called Faddeev-Reshetikhin-Takhtajan construction. In [22, Proposition 4.3], a braided commutative algebra in the category of Yetter–Drinfel’d modules for B(R) was constructed; thus, a bialgebroid was associated to a ﬁnite-dimensional solution R of the quantum Yang–Baxter equation. The construction above of a scalar extension was extended in [12, Theorem 4.6]. Following it, a smash product of a right bialgebroid B over an algebra R, with a braided commutative algebra A in the category of right–right Yetter–Drinfel’d modules for B, is shown to possess a right A-bialgebroid structure. The fundamental importance of scalar extensions from the point of view of Galois extensions by bialgebroids is discussed in Section 3.7. 3.5. The monoidal category of modules An algebra B over a commutative ring k is known to have a k-bialgebra structure if and only if the category of (left or right) B-modules is monoidal such that the forgetful functor to Mk is strict monoidal [35]. A generalization [36, Theorem 5.1] of this fact to bialgebroids is due to Schauenburg. Recall that any (right) module of an R ⊗k Rop -ring (B, s, t) is an R-bimodule via the actions by t(r) and s(r), for r ∈ R. Theorem 3.13. For an algebra R over a commutative ring k, the following data are equivalent on an R ⊗k Rop -ring (B, s, t).

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G. Böhm

(1) A right bialgebroid structure on (B, s, t); (2) A monoidal structure on the category MB of right B-modules such that the forgetful functor MB → R MR is strict monoidal. Applying Theorem 3.13 to the opposite R ⊗k Rop -ring, an analogous equivalence is obtained between left bialgebroid structures and monoidal structures on the category of left modules. At the heart of Theorem 3.13 lies the fact that the right regular B-module is a generator in the category MB . Hence, the B-module structure on the R-module tensor product of any two B-modules can be expressed in terms of the action on 1B ⊗R 1B ∈ B ⊗R B, which deﬁnes a coproduct. In terms of a coproduct b → b(1) ⊗R b(2) := (1B ⊗R 1B )·b, the right B-action in the R-module tensor product of two right B-modules M and N is written as (m ⊗ n) · b = m · b(1) ⊗ n · b(2) , R

R

for m ⊗ n ∈ M ⊗ N, b ∈ B. R

R

(3.9)

A B-module structure on the monoidal unit R is equivalent to a right character : B → R by Lemma 2.5. Recall that, for an R ⊗k Rop -ring (B, s, t), the category MB is isomorphic to the category of algebras for the monad − ⊗R⊗Rop B on R MR (where R-bimodules are considered as right R ⊗k Rop -modules). In light of this fact, Theorem 3.13 is a particular case of a question discussed by Moerdijk in [26]: Having a monad B on a monoidal category M, the monoidal structure of M lifts to a monoidal structure on the category MB of B-algebras (in the sense that the forgetful functor MB → M is strict monoidal, as in part (2) in Theorem 3.13) if and only if B is a monoid in the category of op-monoidal endofunctors on M, that is, a Hopf monad in the terminology of [26] (called a bimonad in [27]). Comparing this result with Theorem 3.13, we obtain other evidence for a characterization of bialgebroids as bimonads as in Theorem 3.5. In the paper [25] by Szlachányi, it was investigated what bialgebroids possess monoidally equivalent module categories. That is, a monoidal Morita theory of bialgebroids was developed. Based on Theorem 3.5, one main result in [25] can be reformulated as in Theorem 3.14 below. Recall that a bimodule, for two monads B : M → M and B : M → M , is a functor M : M → M together with natural transformations : MB → M and λ : B M → M, satisfying the usual compatibility conditions for the right and left actions in a bimodule. If any pair of parallel morphisms in M possesses a coequalizer, and B preserves coequalizers, then the bimodule M induces a functor M : MB → MB , between the categories of algebras for B and B , respectively, with object map (V, v) → Coeq(M(v), V ). If both categories M and M possess coequalizers and both B and B preserve them, then one can deﬁne the inverse of a bimodule as in Morita theory. A B -B bimodule M is said to be the inverse of M provided that M M is naturally equivalent to the identity functor on MB , and M M is naturally equivalent to the identity functor on MB . Theorem 3.14. For two right bialgebroids B and B , over respective base algebras R and R , the right module categories MB and MB are monoidally equivalent if

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195

and only if there exists an invertible bimodule in the 2-category of op-monoidal left adjoint functors, for the monads − ⊗R⊗Rop B : R MR → R MR and − ⊗R ⊗Rop B : R MR → R MR . By standard Morita theory, an equivalence M : MB → MB is of the form M = −⊗B M, for some invertible B-B bimodule M. In [25], monoidality of the equivalence is translated to properties of the Morita equivalence bimodule M. over a commutative ring k were In [24, Deﬁnition 2.1], two algebras R and R √ said to be Morita-equivalent whenever the bimodule categories R MR and R MR are strictly equivalent as k-linear monoidal categories. This property implies that √ op are Morita-equivalent (but not conversely). ⊗k R the algebras R ⊗k Rop and R In this situation, any R ⊗k Rop -ring B (i.e. monoid in the category of R ⊗k Rop ⊗k R op -ring B, with underlying k-algebra B Morita bimodules) determines an R equivalent to B. If B is a right R-bialgebroid, then the forgetful functor MB → R MR is strict monoidal by Theorem 3.13. Hence, the equivalence MB ∼ = M B can be used to induce a monoidal structure on M such that the forgetful functor M MR B B →R is strict monoidal. By Theorem 3.13, we conclude that there is a right R-bialgebroid √ was said to be obtained from B via Moritastructure on B. In [34], the bialgebroid B √ (but not base change. Since Morita equivalent algebras are also Morita-equivalent √ conversely), the construction in Section 3.4.5 is a special instance of a Morita-base change. For a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1, consider B as an R-bimodule (or Rop -bimodule) via right multiplications by the source map s and the target map t. Both B ⊗R B and B ⊗Rop B are left modules for B ⊗k B via the regular actions on the two factors. Associated to B, we construct a category C(B), with two objects ◦ and •. The morphisms with source ◦ are deﬁned to be elements of B⊗R B and the morphisms with source • are elements of B⊗Rop B. The morphisms F , with target ◦ and •, are required to satisfy the following (R-centralizing) conditions (T ◦) and (T •), respectively, for all r ∈ R. (T ◦)

(s(r) ⊗ 1B ) · F = (1B ⊗ t(r)) · F ,

(T •)

(t(r) ⊗ 1B ) · F = (1B ⊗ s(r)) · F .

k

k

k

k

Via composition given by factorwise multiplication, C(B) is a category. The unit morphisms at the objects ◦ and • are 1B ⊗R 1B and 1B ⊗Rop 1B , respectively. The range of the coproduct lies in Hom(◦, ◦) = B×R B, and the range of the co-opposite coproduct cop lies in Hom(•, •) = Bcop ×Rop Bcop . In terms of the category C(B), the deﬁnition of a quasi-triangular bialgebroid as formulated in [3, Proposition 3.13] can be described as follows. Deﬁnition 3.15. For a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1, let C(B) be the category constructed above. An invertible morphism R = R1 ⊗Rop R2 ∈ Hom(•, ◦) (where implicit summation is understood)

196

G. Böhm

is a universal R-matrix provided that for any b ∈ B, the following identity holds in Hom(•, ◦): (b)R = Rcop (b) and (cop ⊗op B)(R) = R R

and

R

(B ⊗op cop )(R) = R R, R

where the (well deﬁned) maps − R : B ⊗op B → B ⊗op B ⊗op B,

b ⊗op b → b ⊗op R1 ⊗op b R2

R − : B ⊗op B → B ⊗op B ⊗op B,

b ⊗op b → bR1 ⊗op R2 ⊗op b

R

R

R

R

R

R

R

R

R

R

R

and

R

are used. A right bialgebroid B with a given universal R-matrix R is called a quasitriangular bialgebroid. The following Theorem 3.16 was obtained in [3, Theorem 3.15], as a generalization of an analogous result for quasi-triangular bialgebras. Theorem 3.16. Consider a quasi-triangular right bialgebroid (B, R) over a base algebra R. The monoidal category of right B-modules is braided, with as braiding the natural isomorphism M ⊗ M → M ⊗ M, R

R

m ⊗ m → m · R2 ⊗ m · R1 . R

R

3.6. The monoidal category of comodules For a bialgebra over a commutative ring k, not only the category of modules but also the category of (left or right) comodules has a monoidal structure such that the forgetful functor to Mk is strict monoidal. In trying to prove an analogue of this result for bialgebroids, the ﬁrst step is to ﬁnd a forgetful functor. A right, say, comodule of (the constituent coring in) an R-bialgebroid is by deﬁnition only a right R-module. To obtain a forgetful functor to the monoidal category of R-bimodules, the following lemma [75, Lemma 1.4.1] is needed. Lemma 3.17. Let R and S be two algebras over a commutative ring k and let C be an S|R-coring. Any right comodule (M, M ) of the R-coring C can be equipped with a unique left S-module structure such that M (m) belongs to the center of the S-bimodule M ⊗R C, for every m ∈ M. This unique left S-action makes M an S-R bimodule. Every C-comodule map becomes an S-R bimodule map. That is, there is a forgetful functor MC → S MR . The left S-action on a right comodule (M, M ) of the R-coring C is constructed as s · m := m[0] · (m[1] · (s ⊗ 1R )), k

for s ∈ S, m ∈ M.

(3.10)

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197

In particular, (3.10) can be used to equip a right comodule of a right R-bialgebroid with an R-R bimodule structure. Applying the construction to co-opposite and opposite bialgebroids, forgetful functors are obtained from categories of left and right comodules of left and right R-bialgebroids to R MR . Theorem 3.18. Let R be an algebra and let B be a right R-bialgebroid. The category MB of right B-comodules is monoidal such that the forgetful functor MB → R MR is strict monoidal. The monoidal unit R in R MR is a right B-comodule via a coaction provided by the source map. One has to verify that, for any two right B-comodules M and N, the diagonal coaction M ⊗ N → M ⊗ N ⊗ B, R

R

R

m ⊗ n → m[0] ⊗ n[0] ⊗ m[1] n[1] R

R

R

(3.11)

is well deﬁned and that the natural coherence isomorphisms in R MR are B-comodule maps. We do not know a converse of Theorem 3.18, that is, an analog of the correspondence (2)⇒(1) in Theorem 3.13 for the category of comodules. A reason for this is that (in contrast to modules of an R-bialgebroid B, which are algebras for the monad − ⊗R⊗Rop B on the monoidal category of R-bimodules) it is not known whether comodules can be described as coalgebras of a comonad on R MR . However, the deﬁnition of a bialgebroid can be dualized in the sense of reversing all arrows in the diagrams in Mk , expressing the axioms of a bialgebroid over a k-algebra. For a ﬂat k-coalgebra C, (e.g. when k is a ﬁeld), C-bicomodules constitute a monoidal category C MC. A monoidal structure is given by cotensor products – a notion dual to a module tensor product. That is, for C-bicomodules M and N, the cotensor product M2C N is the equalizer of the maps M ⊗k N and M ⊗k N , where M is the right coaction on M and N is the left coaction on N. Flatness of the k-module C implies that M2C N is a C-bicomodule via the left C-coaction on M and right C-coaction on N. In this case, dualization of the bialgebroid axioms leads to the notion of a bicoalgebroid over the k-coalgebra C (see [22]). The relation between C-bicoalgebroid structures on a comonoid in C MC and strict monoidal structures on the forgetful functor from its comodule category to C MC is studied in [37] and [38]. Applying Theorem 3.18 to the co-opposite bialgebroid, strict monoidality follows for the forgetful functor B M → Rop MRop , for a right R-bialgebroid B. Applying Theorem 3.18 to opposite bialgebroids, it follows that the forgetful functors B M → B R MR and M → Rop MRop are strict monoidal, for a left R-bialgebroid B. Note that, for an R-bialgebroid B, which is a ﬁnitely generated and projective R-module on the appropriate side, the equivalence in Proposition 2.12 between the categories of comodules for B and modules for its dual is strict (anti)monoidal. The reader should be warned that in the paper [39], a different notion of a comodule is used. For an S|R-coring C, the coproduct and the counit of the R-coring C project to a coproduct and a counit on the quotient R-bimodule C/{(s⊗k 1R )·c−c·(s⊗k 1R ) | c ∈ C, s ∈ S}. Applying the deﬁnition of a comodule of a bimonad in [39, Section 4.1] to a bimonad induced by a right R-bialgebroid B, the resulting notion is a comodule

198

G. Böhm

for the corresponding quotient coring B/{s(r)b − t(r)b | b ∈ B, r ∈ R} (where s and t are the source and target maps of B, respectively). The category of such comodules is not known to be monoidal. 3.7. Algebra extensions by bialgebroids. Galois extensions In analogy with bialgebra extensions, there are two symmetrical notions of an algebra extension by a bialgebroid B. In the action picture, one deals with a B-module algebra M and its invariant subalgebra (with respect to a character deﬁned by the counit). In this picture, Galois property means Galois property of an associated M-ring. Dually, in the coaction picture, one deals with a B-comodule algebra M and its coinvariant subalgebra (with respect to a grouplike element deﬁned by the unit). In this picture, Galois property means Galois property of an associated M-coring. Although the two approaches are symmetric (and equivalent for ﬁnitely generated and projective bialgebroids), the coaction picture is more popular and more developed. We present it in more detail but, for the sake of completeness, we shortly describe the action picture as well. 3.7.1. The action and coaction pictures By Theorem 3.13, the category MB of right modules of a right R-bialgebroid B is monoidal. By deﬁnition, a right B-module algebra is a monoid M in MB . Denote the structure maps of B as in Deﬁnition 3.1. In view of Lemma 2.2, a right B-module algebra is the same as an algebra and right B-module M such that the multiplication in M is R-balanced and (mm ) · b = (m · b(1) )(m · b(2) ) and 1M · b = 1M · s (b) , for m, m ∈ M, b ∈ B, cf. (3.9). Note in passing that, by strict monoidality of the forgetful functor MB → a right B-module algebra M has a canonical R-ring structure. Its unit is the map R → M, r → 1M · s(r) = 1M · t(r). For a right B-module algebra M, B ⊗R M has an M-ring structure. It is called the smash product. The multiplication is (b ⊗R m)(b ⊗R m ) = bb(1) ⊗R (m · b(2) )m , and the unit is m → 1B ⊗R m. The right character on the R-ring (B, s) determines a right character ⊗R M on the M-ring B ⊗R M. Hence, we can consider the invariant subalgebra of the base algebra M, with respect to the right character ⊗R M. It coincides with the -invariants of the (B, s)-module M, R MR ,

N := M = {n ∈ M | n · b = n · s ((b)),

∀b ∈ B}.

In the action picture, the algebra M is said to be a right B-Galois extension of the invariant subalgebra N provided that B ⊗R M is a Galois M-ring with respect to the right character ⊗R M. That is, the canonical map B ⊗ M → N End(M), b ⊗ m → m → (m · b)m R

R

is bijective. Left Galois extensions by a left bialgabroid B are deﬁned symmetrically, referring to a left B-module algebra and its -invariant subalgebra.

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By Theorem 3.18, also the category MB of right comodules of a right R-bialgebroid B is monoidal. By deﬁnition, a right B-comodule algebra is a monoid in MB . In view of Lemma 2.2, a right B-comodule algebra is the same as an algebra and right B-comodule M, with coaction m → m[0] ⊗R m[1] such that the multiplication in M is R-balanced and, for m, m ∈ M, (mm )[0] ⊗ (mm )[1] = m[0] m[0] ⊗ m[1] m[1] R

and

R

1M [0] ⊗ 1M [1] = 1M ⊗ 1B . R

R

(3.12) Note in passing that, by strict monoidality of the forgetful functor MB → R MR , a right B-comodule algebra M has a canonical R-ring structure. Its unit is the map R → M, r → 1M · r = r · 1M . For a right B-comodule algebra M, M ⊗R B has an M-coring structure with left and right M-actions m1 · (m ⊗ b) · m2 = m1 mm2 [0] ⊗ bm2 [1] , R

R

for m1 , m2 ∈ M, m ⊗ b ∈ M ⊗ B, R

R

(3.13) comultiplication m ⊗R b → (m ⊗R b(1) ) ⊗M (1M ⊗R b(2) ), and counit m ⊗R b → m · (b). The grouplike element 1B in the R-coring (B, , ) determines a grouplike element 1M ⊗R 1B in the M-coring M ⊗R B. Hence, we can consider the coinvariant subalgebra of the base algebra M, with respect to the grouplike element 1M ⊗R 1B . It coincides with the 1B -coinvariants of the B-comodule M, N := M 1B = {n ∈ M | n[0] ⊗ n[1] = n ⊗ 1B }. R

R

Note that by right R-linearity of the B-coaction on M and (3.12), for n ∈ N and r ∈ R, (n · r)[0] ⊗ (n · r)[1] = n ⊗ s(r) = (r · n)[0] ⊗ (r · n)[1] . R

R

R

Hence, for n ∈ N and r ∈ R, n(1M · r) = n · r = r · n = (1M · r)n.

(3.14)

In the coaction picture, the algebra M is said to be a right B-Galois extension of the coinvariant subalgebra N provided that M ⊗R B is a Galois M-coring with respect to the grouplike element 1M ⊗R 1B . That is, the canonical map can : M ⊗ M → M ⊗ B, N

R

m ⊗ m → mm[0] ⊗ m[1] N

R

(3.15)

is bijective. Since for a right R-bialgebroid B also the category of left comodules is monoidal, there is a symmetrical notion of a left B-Galois extension N ⊆ M. It is a left B-comodule algebra M, with coinvariant subalgebra N, such that an associated M-coring B ⊗R M is a Galois coring, with respect to the grouplike element 1B ⊗R 1M . Left and right Galois extensions by left bialgebroids are treated symmetrically. For a right comodule algebra M of a right R-bialgebroid B, a right–right relative Hopf module is a right M-module in MB . The category of right–right relative Hopf

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modules is denoted by MB M , and it turns out to be isomorphic to the category of right comodules for the M-coring (3.13). Hence, the grouplike element 1M ⊗R 1B ∈ M⊗R B determines an adjunction as in (2.6) between MB M and the category MN of right modules for the coinvariant subalgebra N of M. It will be denoted as − ⊗ M : MN → MB M N

and

(−)coB : MB M → MN .

(3.16)

Recall from Section 2.2 that for a B-Galois extension N ⊆ M, this adjunction is interesting from the descent theory point of view. For a ﬁnitely generated and projective bialgebroid, the action and coaction pictures are equivalent in the sense of Proposition 3.19. This equivalence was observed (in a slightly more restricted context) in [12, Theorem and Deﬁnition 3.3]. Proposition 3.19. Let B be a right R-bialgebroid, which is a ﬁnitely generated and projective right R-module via right multiplication by the source map. (1) There is a bijective correspondence between right B-module algebra structures and right (B ∗ )op -comodule algebra structures on a given algebra M. (2) The invariant subalgebra N of a right B-module algebra M (with respect to the right character given by the counit) is the same as the coinvariant subalgebra of the corresponding right (B ∗ )op -comodule algebra M (with respect to the grouplike element given by the unit). (3) A right B-module algebra M is a B-Galois extension of its invariant subalgebra N in the action picture if and only if M is a (B∗ )op -Galois extension of N in the coaction picture. ∼ Part (1) of Proposition 3.19 follows by the strict monoidal equivalence MB = ∗ op M(B ) . Parts (2) and (3) follow by Proposition 2.13 since the M-ring B ⊗R M, associated to a right B-module algebra M, is the left M-dual of the M-coring M ⊗R (B∗ )op , associated to the right (B ∗ )op -comodule algebra M. Consider a right R-bialgebroid B, which is a ﬁnitely generated and projective right R-module via the source map. Then, for a right B-module algebra M, the category of right–right (M, (B∗ )op ) relative Hopf modules is equivalent also to the category of right modules for the smash product algebra B ⊗R M. In the remainder of these notes, only the coaction picture of Galois extensions will be used. 3.7.2. Quantum torsors and bi-Galois extensions Following the work of Grunspan and Schauenburg [40–43], for a bialgebra B over a commutative ring k, a faithfully ﬂat right B-Galois extension T of k can be described without explicit mention of the bialgebra B. Instead, a quantum torsor structure is introduced on T, from which B can be reconstructed uniquely. What is more, a quantum torsor determines a second k-bialgebra B for which T is a left B -Galois extension of k. It is said that any faithfully ﬂat Galois extension of k by a k-bialgebra is in fact a bi-Galois extension. The categories of (left) comodules for the bialgebras B and B are monoidally equivalent.

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Such a description of faithfully ﬂat Galois extensions by bialgebroids was developed in the PhD thesis of Hobst [44] and in the paper [45]. Deﬁnition 3.20. For two algebras R and S over a commutative ring k, an R-S torsor is a pair (T, τ). Here, T is an R ⊗k S-ring with underlying k-algebra T and unit maps α : R → T and β : S → T (with commuting ranges in T). Considering T as an R-S bimodule and as an S-R bimodule via the maps α and β, τ is an S-R bimodule map T → T ⊗R T ⊗S T , t → t 1 ⊗R t 2 ⊗S t 3 (where implicit summation is understood), satisfying the following axioms, for t, t ∈ T , r ∈ R, and s ∈ S. (i) (τ ⊗R T ⊗S T ) ◦ τ = (T ⊗R T ⊗S τ) ◦ τ (coassociativity), (ii) (μR ⊗S T) ◦ τ = β ⊗S T and (T ⊗R μS ) ◦ τ = T ⊗R α (left and right counitality), (iii) τ(1T ) = 1T ⊗R 1T ⊗S 1T (unitality), (iv) α(r)t 1 ⊗R t 2 ⊗S t 3 = t 1 ⊗R t 2 α(r) ⊗S t 3 and t 1 ⊗R β(s)t 2 ⊗S t 3 = t 1 ⊗R t 2 ⊗S t 3 β(s), (centrality conditions) (v) τ(tt ) = t 1 t 1 ⊗R t 2 t 2 ⊗S t 3 t 3 (multiplicativity), where μR and μS denote multiplication in the R-ring (T, α) and the S-ring (T, β), respectively. An R-S torsor (T, τ) is said to be faithfully ﬂat if T is a faithfully ﬂat right R-module and a faithfully ﬂat left S-module. Note that axiom (iv) in Deﬁnition 3.20 is needed in order for the multiplication in axiom (v) to be well deﬁned. Theorem 3.21. For two k-algebras R and S, there is a bijective correspondence between the following sets of data. (i) Faithfully ﬂat R-S torsors (T, τ). (ii) Right R-bialgebroids B and left faithfully ﬂat right B-Galois extensions S ⊆ T such that T is a right faithfully ﬂat R-ring. (iii) Left S-bialgebroids B and right faithfully ﬂat left B -Galois extensions R ⊆ T such that T is a left faithfully ﬂat S-ring. Furthermore, a faithfully ﬂat R-S torsor T is a B -B bicomodule, that is, the left B , and right B-coactions on T do commute. Starting with the data in part (ii) of Theorem 3.21, a torsor map on T is constructed in terms of the B-coaction T : T → T ⊗R B, and the inverse of the canonical map (3.15) (with the role of the comodule algebra M in (3.15) played by T ), as τ := (T ⊗R can−1 (1T ⊗R −)) ◦ T . Conversely, to a faithfully ﬂat R-S torsor (T, τ) (with multiplication μS in the S-ring (T, β)), one associates a right R-bialgebroid B deﬁned on the R-R bimodule given by the equalizer of the maps (μS ⊗R T ⊗S T ) ◦ (T ⊗S τ) and α ⊗R T ⊗S T : T ⊗S T → T ⊗R T ⊗S T .

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Theorem 3.22. For two k-algebras R and S, consider a faithfully ﬂat R-S torsor (T, τ). Let B and B be the associated bialgebroids in Theorem 3.21. Assume that T is a faithfully ﬂat right S-module and B is a ﬂat right S-module. Then the categories of left B- and B -comodules are monoidally equivalent. Note that the assumptions made about the right S-modules T and B in Theorem 3.22 become redundant when working with one commutative base ring R = S and equal unit maps α = β. The equivalence in Theorem 3.22 is given by T 2B − : B M → B M, a cotensor product with the B -B bicomodule T . (Recall that the notion of a cotensor product is dual to the one of module tensor product. That is, for a right B-comodule (M, M ) and a left B-comodule (N, N ), M2B N is the equalizer of the maps M ⊗R N and M ⊗R N .) 3.7.3. Galois extensions by ﬁnitely generated and projective bialgebroids The bialgebra B, for which a given algebra extension is B-Galois, is nonunique. Obviously, there are (potentially) even more possibilities for the choice of B if it is allowed to be a bialgebroid. Still, as a main advantage of studying Galois extensions by bialgebroids, in appropriate ﬁnite cases all possible bialgebroids B can be related to a canonical one. The following Theorem 3.24 is a mild generalization of [12, Proposition 4.12]. Deﬁnition 3.23. Let B be a right bialgebroid over an algebra R. A right–right Yetter–Drinfel’d module for B is a right B-module and right B-comodule M (with one and the same underlying R-bimodule structure) such that the following compatibility condition holds, (m · b(2) )[0] ⊗ b(1) (m · b(2) )[1] = m[0] · b(1) ⊗ m[1] b(2) , R

R

for m ∈ M, b ∈ B.

It follows by a symmetrical version of [33, Proposition 4.4] that the category of right–right Yetter–Drinfel’d modules of a right bialgebroid B is isomorphic to the weak center of the monoidal category of right B-modules. Hence, it is monoidal and prebraided. Following the paper [12], the construction in Section 3.4.7 can be extended to a braided commutative algebra A in the category of right–right Yetter– Drinfel’d modules of a right R-bialgebroid B. That is, the smash product algebra A#B can be proven to carry the structure of a right A-bialgebroid, which is called a scalar extension of B by A. In Theorem 3.24, the center of a bimodule M of an algebra R is denoted by M R . Theorem 3.24. For an algebra R, consider a right R-bialgebroid B, which is a ﬁnitely generated and projective left R-module via right multiplication by the target map. Let N ⊆ M be a right B-Galois extension. Then N ⊆ M is a right Galois extension by a right bialgebroid (M ⊗N M)N over the base algebra M N . What is more, the bialgebroid (M ⊗N M)N is isomorphic to a scalar extension of B. In proving Theorem 3.24, the following key ideas are used. First, a braided commutative algebra structure, in the category of right–right Yetter–Drinfel’d modules

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for B, is constructed on M N . The B-coaction on M N is given by restriction of the B-coaction on M. The B-action on M N is of the Miyashita–Ulbrich type, that is, it is given in terms of the inverse of the canonical map (3.15). Introducing the index notation can−1 (1M ⊗R b) = b{1} ⊗N b{2} , for b ∈ B (implicit summation is understood), the right B-action on M N is a · b := b{1} ab{2} , for b ∈ B and a ∈ M N . Since in this way M N is a braided commutative algebra in the category of right– right Yetter–Drinfel’d modules for B, there exists a right M N -bialgebroid M N ⊗R B (cf. Section 3.4.7). Restriction of the B-canonical map (3.15) establishes a bijection (M ⊗N M)N → M N ⊗R B. Hence, it induces an M N -bialgebroid structure on (M ⊗N M)N and also an (M ⊗N M)N -comodule algebra structure on M. After checking that the coinvariants of the (M ⊗N M)N -comodule M are precisely the elements of N, the (M ⊗N M)N -Galois property of the extension N ⊆ M becomes obvious: the (M ⊗N M)N -canonical map differs from the B-canonical map (3.15) by an isomorphism. 3.7.4. Depth 2 algebra extensions Classical ﬁnitary Galois extensions of ﬁelds can be characterized inherently by normality and separability properties, without referring to the Galois group G. That is, a (unique up to isomorphism) ﬁnite Galois group G := Aut K (F ) is determined by any normal and separable ﬁeld extension F of K. While no such inherent characterization of Galois extensions by (ﬁnitely generated and projective) bialgebras is known, a most important achievement in the Galois theory of bialgebroids is a characterization of Galois extensions by ﬁnitely generated and projective bialgebroids. A ﬁrst result in this direction was given in [12, Theorem 3.7]. At the level of generality presented here, it was proven in [13, Theorem 2.1]. The following deﬁnition in [23, Deﬁnition 3.1] was abstracted from depth 2 extensions of C∗ -algebras. Deﬁnition 3.25. Consider an extension N ⊆ M of algebras. It is said to satisfy the right (resp. left) depth 2 condition if the M-N bimodule (resp. N-M bimodule) M ⊗N M is a direct summand in a ﬁnite direct sum of copies of M. Note that the right depth 2 property of an algebra extension N ⊆ M is equivalent to the existence of ﬁnitely many elements γk ∈ N End N (M) ∼ = M HomN (M ⊗N M, M) and ck ∈ (M ⊗N M)N ∼ = M HomN (M, M ⊗N M), the so called right depth 2 quasibasis satisfying the identity mγk (m )ck = m ⊗ m for m, m ∈ M. (3.17) k

N

Deﬁnition 3.26. An extension N ⊆ M of algebras is balanced if all endomorphisms of M, as a left module for the algebra E := EndN (M), are given by right multiplication by some element of N. Theorem 3.27. For an algebra extension N ⊆ M, the following properties are equivalent.

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(i) N ⊆ M is a right Galois extension by some right R-bialgebroid B, which is a ﬁnitely generated and projective left R-module via right multiplication by the target map. (ii) The algebra extension N ⊆ M is balanced and satisﬁes the right depth 2 condition. If N ⊆ M is a right Galois extension by a right R-bialgebroid B, then M ⊗N M ∼ = M ⊗R B as M-N bimodules. Hence, the right depth 2 condition follows by ﬁnitely generated projectivity of the left R-module B. The map, given by left multiplication by an element m ∈ M, and the map, given by right multiplication by r ∈ R, are both right N-linear. Therefore, a left E = End N (M)-module endomorphism of M is given by right multiplication by an element x ∈ M R . Since by (3.14) also the right action (2.7) on M by φ ∈ ∗ B is a right N-module map, it follows that x[0] φ(x[1] ) = xφ(1B ) for all φ ∈ ∗ B. Together with the ﬁnitely generated projectivity of the left R-module B, this implies that x belongs to the coinvariant subalgebra N, and hence the extension N ⊆ M is balanced. By Theorem 3.24, if N ⊆ M is a right Galois extension by some ﬁnitely generated projective right R-bialgebroid B, then it is a Galois extension by the (ﬁnitely generated projective) right M N -bialgebroid (M ⊗N M)N . Hence, in the converse direction, a balanced algebra extension N ⊆ M, satisfying the right depth 2 condition, is shown to be a right Galois extension by a right M N -bialgebroid (M ⊗N M)N , as constructed in [23, Section 5]. (In fact, in [23], both the left and right depth 2 properties are assumed. It is proven in [13] that the construction works for one-sided depth 2 extensions as well.) The coproduct in (M ⊗N M)N and its coaction on M are constructed in terms of the right depth 2 quasi-basis (3.17). Let us mention that the only point in the proof where the balanced property is used is to show that the (M ⊗N M)N -coinvariants in M are precisely the elements of N. 4. Hopf algebroids A Hopf algebra is a bialgebra H equipped with an additional antipode map H → H. The antipode is known to be a bialgebra map from H to the opposite co-opposite of H. It does not seem to be possible to deﬁne a Hopf algebroid based on this analogy. Startop ing with a, say left, bialgebroid H, its opposite co-opposite Hcop is a right bialgebroid. op There is no sensible notion of a bialgebroid map H → Hcop . If we choose as a guiding op principle the antipode of a Hopf algebroid H to be a bialgebroid map H → Hcop , op then H and Hcop need to carry the same, say left, bialgebroid structure. This means that the underlying algebra H must be equipped with both left and right bialgebroid structures. The ﬁrst deﬁnition fulﬁlling this requirement was proposed in [46, Deﬁnition 4.1], where, however, the antipode was deﬁned to be bijective. Bijectivity of the antipode was relaxed in [47, Deﬁnition 2.2]. Here, we present a set of axioms, which is equivalent to [47, Deﬁnition 2.2], as it was formulated in [48, Remark 2.1]. Deﬁnition 4.1. For two algebras R and L over a commutative ring k, a Hopf algebroid over the base algebras R and L is a triple H = (HL , HR , S). Here HL is a left

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L-bialgebroid and HR is a right R-bialgebroid such that their underlying k-algebra H is the same. The antipode S is a k-module map H → H. Denote the R ⊗k Rop -ring structure of HR by (H, sR , tR ) and its R-coring structure by (H, R , R ). Similarly, denote the L ⊗k Lop -ring structure of HL by (H, sL , tL ) and its L-coring structure by (H, L , L ). Denote the multiplication in the R-ring (H, sR ) by μR and denote the multiplication in the L-ring (H, sL ) by μL . These structures are subject to the following compatibility axioms. (i) (ii) (iii) (iv)

sL ◦L ◦tR = tR , tL ◦L ◦sR = sR , sR ◦R ◦tL = tL , and tR ◦R ◦sL = sL . (L ⊗R H)◦R = (H ⊗L R )◦L and (R ⊗L H)◦L = (H ⊗R L )◦R . For l ∈ L, r ∈ R and h ∈ H, S(tL (l)htR (r)) = sR (r)S(h)sL (l). μL ◦ (S ⊗L H) ◦ L = sR ◦ R and μR ◦ (H ⊗R S) ◦ R = sL ◦ L .

Remark 4.2. The Hopf algebroid axioms in Deﬁnition 4.1 require some interpretation. (1) By the bialgebroid axioms, all the maps sL ◦ L , tL ◦ L , sR ◦ R , and tR ◦ R are idempotent maps H → H. Hence, the message of axiom (i) is that the ranges of sL and tR and the ranges of sR and tL are coinciding subalgebras of H. These axioms imply that the coproduct L in HL is not only an L-bimodule map but also an R-bimodule map. Symmetrically, R is an L-bimodule map so that axiom (ii) makes sense. (2) The k-module H underlying a (left or right) bialgebroid is a left and right comodule via the coproduct. Hence, the k-module H underlying a Hopf algebroid is a left and right comodule for the constituent left and right bialgebroids HL and HR , via the two coproducts L and R . Axiom (ii) expresses the property that these regular coactions commute, that is, H is an HL -HR bicomodule and also an HR -HL bicomodule. Alternatively, considering H and H ⊗L H as right HR -comodules via the respective coactions R and H ⊗L R , the ﬁrst axiom in (ii) expresses that L is a right HR -comodule map. Symmetrically, this condition can be interpreted as a left HL -comodule map property of R . Similarly, the second axiom in (ii) can be read as right HL -colinearity of R or left HR -colinearity of L . (3) Axiom (iii) formulates the R-L bimodule map property of the antipode, which is needed in order for axiom (iv) to make sense. (4) In analogy with the Hopf algebra axioms, axiom (iv) says that the antipode is a convolution inverse of the identity map of H in some generalized sense. The notion of convolution products in the case of two different base algebras L and R is discussed in Section 4.5.2. Since in a Hopf algebroid H, there are two constituent bialgebroids HL and HR , in these notes we use two versions of Sweedler’s index notation in parallel to denote components of the coproducts L and R . We will use lower indices in the case of a left bialgebroid HL , that is, we write L (h) = h(1) ⊗L h(2) , and upper indices in the case of a right bialgebroid HR , that is, we write R (h) = h(1) ⊗R h(2) , for h ∈ H, where implicit summation is understood in both cases. Analogously, we use

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upper indices to denote components of a coaction by HR and lower indices to denote components of a coaction by HL . 4.1. Examples and constructions Before turning to a study of the structure of Hopf algebroids, let us present some examples. 4.1.1. Hopf algebras A Hopf algebra H over a commutative ring k is an example of a Hopf algebroid over base algebras R = k = L. Both bialgebroids HL and HR are equal to the k-bialgebra H, and the Hopf algebra antipode of H satisﬁes the Hopf algebroid axioms. Certainly, not every Hopf algebroid over base algebras R = k = L is a Hopf algebra (see e.g. Section 4.1.4). Examples of this kind have been constructed in [49], as follows. Let H be a Hopf algebra over k, with coproduct L : h → h(1) ⊗k h(2) , counit L , and antipode S. Let χ be a character on H, that is, a k-algebra map H → k. The coproduct R : h → h(1) ⊗k χ S(h(2) ) h(3) and the counit R := χ deﬁne a second bialgebra structure on the k-algebra H. Looking at these two bialgebras as left and right k-bialgebroids respectively, we obtain a Hopf algebroid with a twisted antipode h → χ(h(1) )S(h(2) ). This construction was extended in [50, Theorem 4.2], where new examples of Hopf algebroids were constructed by twisting a (bijective) antipode of a given Hopf algebroid. 4.1.2. Weak Hopf algebras A weak Hopf algebra over a commutative ring k is a weak bialgebra H equipped with a k-linear antipode map S : H → H, subject to the following axioms [29]. For h ∈ H, h(1) S(h(2) ) = L (h),

S(h(1) )h(2) = R (h), and

S(h(1) )h(2) S(h(3) ) = S(h),

where the maps L and R were introduced in Section 3.2.2. The right R-bialgebroid and the left Rop -bialgebroid constructed for a weak Hopf algebra H in Section 3.2.2, together with the antipode S, satisfy the Hopf algebroid axioms. In particular, consider a small groupoid with ﬁnitely many objects. By Section 3.2.2, the free k-module spanned by its morphisms is a weak k-bialgebra. It can be equipped with an antipode by putting S(f ) := f −1 for every morphism f and extending it k-linearly. Motivated by this example, weak Hopf algebras and sometimes also Hopf algebroids are called quantum groupoids in the literature. Weak Hopf algebras have a nice and well-understood representation theory. The category of ﬁnite-dimensional modules of a ﬁnite-dimensional semisimple weak Hopf algebra H over a ﬁeld k is a k-linear semisimple category with ﬁnitely many inequivalent irreducible objects, with all hom spaces ﬁnite dimensional. It is a monoidal category with left and right duals. A category with the listed properties is termed a fusion category. Conversely, based on Tannaka–Krein type reconstruction theorems

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in [32] and [51], it was proven in [52] that any fusion category is equivalent to the category of ﬁnite-dimensional modules of a (nonunique) ﬁnite-dimensional semisimple weak Hopf algebra. 4.1.3. R ⊗ Rop The left and right bialgebroids on an algebra of the form R ⊗k Rop , constructed for any k-algebra R in Section 3.2.3, form a Hopf algebroid together with the antipode r ⊗k r → r ⊗k r. 4.1.4. The algebraic quantum torus Consider an algebra Tq over a commutative ring k generated by two invertible elements U and V subject to the relation UV = qVU, where q is an invertible element in k. Tq possesses a right bialgebroid structure over the commutative subalgebra R generated by U. Both the source and target maps are given by the inclusion R → Tq . The coproduct and counit are deﬁned by R : V m U n → V m U n ⊗R V m and R : V m U n → U n , respectively. Symmetrically, there is a left R-bialgebroid structure given by the coproduct L : U n V m → U n V m ⊗R V m and counit L : U n V m → U n . Together with the antipode S : U n V m → V −m U n , they constitute a Hopf algebroid. 4.1.5. Scalar extension Consider a Hopf algebra H and a braided commutative algebra A in the category of right–right Yetter–Drinfel’d modules of H. As was seen in Section 3.4.7, the smash product algebra A#H carries a right A-bialgebroid structure. If the antipode S of H is bijective, then the A-bialgebroid structure of A#H extends to a Hopf algebroid. Indeed, A#H is a left Aop -bialgebroid via the a#h → source map a → a[0] · S(a[1] )#1B , target map a → a[0] #a[1] , coproduct (a#h(1) ) ⊗Aop (1A #h(2) ), and counit a#h → a[0] · S −1 hS −1 (a[1] ) . The (bijective) antipode is given by a#h → a[0] · S(h(2) )#a[1] S(h(1) ). 4.2. Basic properties of Hopf algebroids In this section, some consequences of Deﬁnition 4.1 of a Hopf algebroid are recalled from [47, Section 2]. The opposite co-opposite of a Hopf algebra H is a Hopf algebra, with the same antipode S of H. If S is bijective, then also the opposite and the co-opposite of H are Hopf algebras, with antipode S −1 . A generalization of these facts to Hopf algebroids is given below. Proposition 4.3. Consider a Hopf algebroid (HL , HR , S) over base algebras L and R. Then op

op

(1) The triple ((HR )cop , (HL )cop , S) is a Hopf algebroid over the base algebras Rop and Lop . (2) If the antipode S is bijective, then ((HR )op , (HL )op , S −1 ) is a Hopf algebroid over the base algebras R and L and ((HL )cop , (HR )cop , S −1 ) is a Hopf algebroid over the base algebras Lop and Rop .

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Proposition 4.4 states the expected behavior of the antipode of a Hopf algebroid with respect to the underlying ring and coring structures. Consider a Hopf algebroid H over base algebras L and R, with structure maps denoted as in Deﬁnition 4.1. It follows immediately by axiom (i) in Deﬁnition 4.1 that the base algebras L and R are anti-isomorphic. Indeed, there are inverse algebra isomorphisms L ◦ sR : Rop → L

and

R ◦ tL : L → Rop .

(4.1)

Symmetrically, there are inverse algebra isomorphisms R ◦ sL : Lop → R

and L ◦ tR : R → Lop .

(4.2)

Proposition 4.4. Let H be a Hopf algebroid over base algebras L and R with structure maps denoted as in Deﬁnition 4.1. The following assertions hold. (1) The antipode S is a homomorphism of R ⊗k Rop -rings (H, sR , tR ) → (Hop , sL ◦ (L ◦ sR ), tL ◦ (L ◦ sR )) and also a homomorphism of L ⊗k Lop -rings (H, sL , tL ) → (Hop , sR ◦ (R ◦ sL ), tR ◦ (R ◦ sL )). In particular, S is a k-algebra antihomomorphism H → H. (2) The antipode S is a homomorphism of R-corings cop

(H, R , R ) → (H, L , (R ◦ sL ) ◦ L ), cop where L is considered as a map H → H ⊗Lop H ∼ = H ⊗R H, via the isomorphism induced by the algebra isomorphism (4.2). Symmetrically, S is a homomorphism of L-corings cop

(H, L , L ) → (H, R , (L ◦ sR ) ◦ R ), where R is considered as a map H → H ⊗Rop H ∼ = H ⊗L H, via the isomorphism induced by the algebra isomorphism (4.1). cop

The k-dual H ∗ of a ﬁnitely generated and projective Hopf algebra H over a commutative ring k is a Hopf algebra. The antipode in H ∗ is the transpose of the antipode of H. No generalization of this fact for Hopf algebroids is known. Although the dual H∗ of a ﬁnitely generated and projective Hopf algebroid H has a bialgebroid structure (cf. Section 3.3), the transpose of the antipode in H is not an endomorphism of H∗ . Only duals of Frobenius Hopf algebroids are known to be (Frobenius) Hopf algebroids (see Section 4.4.2). 4.3. Comodules of Hopf algebroids In a Hopf algebroid, the constituent left and right bialgebroids are deﬁned on the same underlying algebra. Therefore, modules for the two bialgebroids coincide. This

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is not the case with comodules: the two bialgebroids have different underlying corings (over anti-isomorphic base algebras, cf. (4.1) and (4.2)), which have a priori different categories of (say, right) comodules. We take the opportunity here to call the reader’s attention to a regrettable error in the literature. Based on [53, Theorem 2.6], whose proof turned out to be incorrect, the categories of (right) comodules of two constituent bialgebroids in a Hopf algebroid were claimed to be strict monoidally isomorphic in [48, Theorem 2.2]. Since it turned out recently that in [53, Theorem 2.6] some assumptions are missing, the derived result [48, Theorem 2.2] need not hold either at the stated level of generality. (There is a similar error in [54, Proposition 3.1].). Regrettably, this error inﬂuences some results also in [55], [48], and [56]. In the current section and in 4.5, we present the corrected statements. 4.3.1. Comodules of a Hopf algebroid and of its constituent bialgebroids Since, as was explained in the ﬁrst paragraph of Section 4.3, comodules of the two constituent bialgebroids in a Hopf algebroid are in general different notions, none of them can be expected to be a well-working deﬁnition of a comodule of a Hopf-algebroid. The following deﬁnition of a comodule for a Hopf algebroid, as a compatible comodule of both constituent bialgebroids, was suggested in [54, Deﬁnition 3.2] and [12, Section 2.2]. Deﬁnition 4.5. Let H = (HL , HR , S) be a Hopf algebroid over base k-algebras L and R with structure maps denoted as in Deﬁnition 4.1. A right H-comodule is a right L-module as well as a right R-module M, together with a right HR -coaction R : M → M ⊗R H and a right HL -coaction L : M → M ⊗L H such that R is an HL -comodule map and L is an HR -comodule map. Explicitly, R : m → m[0] ⊗R m[1] is a right L-module map in the sense that (m · l)[0] ⊗ (m · l)[1] = m[0] ⊗ tL (l)m[1] , R

R

for m ∈ M, l ∈ L,

L : m → m[0] ⊗L m[1] is a right R-module map in the sense that (m · r)[0] ⊗ (m · r)[1] = m[0] ⊗ m[1] sR (r), L

L

for m ∈ M, r ∈ R,

and the following compatibility conditions hold. (R ⊗ H) ◦ L = (M ⊗ L ) ◦ R L

R

and (L ⊗ H) ◦ R = (M ⊗ R ) ◦ L . R

L

Morphisms of H-comodules are precisely those morphism that are HL - and HR -comodule maps. The category of right comodules of a Hopf algebroid H is denoted by MH . It is not difﬁcult to see that the right R- and L-actions on a right H-comodule M necessarily commute. That is, M carries the structure of a right L ⊗k R-module. Note that by Deﬁnition 4.1 (i) and (ii), the right R⊗k L-module H, with R-action via the right source map sR and L-action via the left target map tL , is a right comodule of the Hopf algebroid H, via the two coactions given by the two coproducts R and L . Left comodules of a Hopf algebroid H are deﬁned symmetrically, and their category is denoted by H M.

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Remark 4.6. The antipode S in a Hopf algebroid H deﬁnes a functor MH → H M. Indeed, if M is a right H-comodule, with HR -coaction m → m[0] ⊗R m[1] and HL -coaction m → m[0] ⊗L m[1] , then it is a left H-comodule with left R-action r · m = m · L (tR (r)), left L-action l · m = m · R (tL (l)) (where the notations in Deﬁnition 4.1 are used), and the respective coactions m → S(m[1] ) ⊗ m[0] R

and m → S(m[1] ) ⊗ m[0] . L

Right H-comodule maps are also left H-comodule maps for these coactions. A functor H M → MH is constructed symmetrically. Although comodules of a Hopf algebroid H = (HL , HR , S) cannot be described as comodules of a coring, the free-forgetful adjunction (cf. [15, 18.13(2)]), corresponding to the L-coring underlying HL , lifts to an adjunction between the categories MH and ML . Indeed, the forgetful functor MH → ML has a right adjoint − ⊗L H : ML → MH . The unit and counit of the adjunction are given for a right H-comodule (M, L , R ) and a right L-module N by the maps L : M → M ⊗ H L

and

N ⊗ L : N ⊗ H → N, L

L

(4.3)

respectively, where L is the counit of HL . There is a similar adjunction between the categories MH and MR . Our next task is to look for situations when the category of comodules of a Hopf algebroid coincides with the comodule category of one the constituent bialgebroids. Recall from Remark 4.2 (2) that for a Hopf algebroid H = (HL , HR , S), the underlying k-module H is an HL -HR bicomodule and an HR -HL bicomodule via the coactions given by the coproducts. In appropriate situations, taking cotensor products with a bicomodule induces a functor between the comodule categories of the two corings (see [15, 22.3] and its Erratum). In Theorem 4.7, functors of this type are considered. Recall from Section 3.6 that any right comodule of a right bialgebroid over a k-algebra R possesses a unique R-bimodule structure such that any comodule map is R-bilinear. Thus, if H is a Hopf algebroid over the anti-isomorphic base k-algebras L and R, then any right comodule of the constituent right (or left) bialgebroid can be regarded as a right L ⊗k R-module and the coaction is a right R ⊗k L-module map. Theorem 4.7. Let H = (HL , HR , S) be a Hopf algebroid over base k-algebras L and R, with structure maps denoted as in Deﬁnition 4.1. Consider the R- and L-actions on H that deﬁne its coring structures (cf. Deﬁnitions 3.1 and 3.3). If the equalizer M

/ M⊗H R

⊗R H M⊗R R

// M ⊗ H ⊗ H R

R

(4.4)

in the category ML of right L-modules is preserved by both functors − ⊗L H ⊗L H and − ⊗L H ⊗R H : ML → Mk , for any right HR -comodule (M, ), then the forgetful functor MH → MHR is an isomorphism.

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By standard terminology, the conditions in Theorem 4.7 are phrased as the equalizer (4.4) in ML is H ⊗L H-pure and H ⊗R H-pure. Symmetrical conditions imply that the forgetful functor to the category of right HL -comodules is an isomorphism. Theorem 4.7 is proved by constructing the inverse of the forgetful functor, that is, by equipping any right HR -comodule with an H-comodule structure. Any right HR -comodule M is isomorphic, as a right R-module, to the cotensor product M2H of M with the left HR comodule H. Since under the assumptions in Theorem 4.7, M2H is a right HL -comodule via M2L : M2H → M2(H ⊗L H) ∼ = (M2H) ⊗L H, the isomorphism M ∼ = M2H induces a right HL -coaction on M. Moreover, by the assumptions in Theorem 4.7, we have isomorphisms M2(H ⊗L H) ∼ = M ⊗L H, M2(H ⊗R H) ∼ = M ⊗R H, M2(H ⊗R H ⊗L H) ∼ = M ⊗R H ⊗L H, and M2(H ⊗L H ⊗R H) ∼ = M ⊗L H ⊗R H. The compatibility conditions between the HR - and HL -coactions follow by these isomorphisms, the Hopf algebroid axioms of Deﬁnition 4.1 (ii), and functoriality of the cotensor product. With similar methods, any HR -comodule map is checked to be also HL -colinear. The purity conditions in Theorem 4.7 are checked to hold in all the examples in Section 4.1. Moreover, if a Hopf algebroid is a ﬂat left R-module (via right multiplication by the target map of the right bialgebroid) and a ﬂat left L-module (via left multiplication by the source map of the left bialgebroid) then it satisﬁes all purity conditions in Theorem 4.7 since taking tensor products with ﬂat modules preserves any equalizer. 4.3.2. Coinvariants in a comodule of a Hopf algebroid By Deﬁnition 4.5, a comodule M of a Hopf algebroid H = (HL , HR , S) is a comodule for both constituent bialgebroids HL and HR . Since the unit element 1H is grouplike for both corings underlying HL and HR , one can speak about coinvariants M coHR = {m ∈ M | R (M) = m ⊗ 1H } R

of M with respect to the HR -coaction R or coinvariants M coHL = {m ∈ M | L (M) = m ⊗ 1H } L

with respect to the HL -coaction L . Proposition 4.8 relates these two notions. It is of crucial importance from the point of view of Galois theory (see Section 4.5). Proposition 4.8. Let H = (HL , HR , S) be a Hopf algebroid and (M, L , R ) be a right H-comodule. Then, any coinvariant of the HR -comodule (M, R ) is coinvariant also for the HL -comodule (M, L ). If moreover the antipode S is bijective, then the coinvariants of the HR -comodule (M, R ) and the HL -comodule (M, L ) coincide. For a right H-comodule (M, L , R ), consider the map M : M ⊗ H → M ⊗ H, R

L

m ⊗R h → L (m) · S(h),

(4.5)

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where (using the notations in Deﬁnition 4.1) H is a left L-module via the source map sL and a left R-module via the target map tR , and M ⊗L H is understood to be a right H-module via the second factor. Since M (R (m)) = m ⊗L 1H and M (m ⊗R 1H ) = L (m), we have the ﬁrst claim in Proposition 4.8 proved. In order to prove the second assertion, note that if S is an isomorphism, then so is M , with −1 inverse −1 M (m ⊗L h) = S (h) · R (m), where M ⊗R H is understood to be a left H-module via the second factor. 4.3.3. Comodule algebras of a Hopf algebroid As it was explained in Section 3.7, from the point of view of Galois theory (in the coaction picture), monoidality of the category of comodules is of central importance. Theorem 4.9 replaces the unjustiﬁed theorem [48, Theorem 2.2] (cf. the ﬁrst paragraph of Section 4.3). By Deﬁnition 4.5, a right comodule of a Hopf algebroid H over base k-algebras L and R is a right L ⊗k R-module. Since L and R are anti-isomorphic algebras, we may regard alternatively any H-comodule as an R-bimodule by translating the right L-action to a left R-action via the algebra anti-isomorphism (4.1). Theorem 4.9. For a Hopf algebroid (HL , HR , S) over base k-algebras L and R, the category MH of right H-comodules is monoidal. Moreover, there are strict monoidal forgetful functors, rendering the following diagram commutative. MH

/ MHR

MHL

/ R MR .

Commutativity of the diagram in Theorem 4.9 follows by comparing the unique R-actions that make R-bilinear the HR -coaction and the HL -coaction in an H-comodule. Strict monoidality of the functor on the right-hand side was proved in Theorem 3.18. Strict monoidality of the functor in the bottom row follows by applying Theorem 3.18 to the opposite of the bialgebroid HL and identifying Lop -bimodules and R-bimodules via the algebra isomorphism (4.1). To see strict monoidality of the remaining two functors, recall that by Theorem 3.18 – applied to HR and the opposite of HL – the R-module tensor product of any two H-comodules is an HR -comodule and an HL -comodule, via the diagonal coactions, cf. (3.11). It is straightforward to check compatibility of these coactions in the sense of Deﬁnition 4.5. Similarly, R(∼ = Lop ) is known to be an HR -comodule and an HL -comodule and compatibility of the coactions is obvious. Finally, the R-module tensor product of H-comodule maps is an HR -comodule map and an HL -comodule map by Theorem 3.18. Thus, it is an Hcomodule map. By Theorem 3.18 also the natural coherence transformations in R MR are HR - and HL -comodule maps, so H-comodule maps, which proves Theorem 4.9. Theorem 4.9 enables us to introduce comodule algebras of Hopf algebroids. Deﬁnition 4.10. A right comodule algebra of a Hopf algebroid H is a monoid in the monoidal category MH of right H-comodules, explicitly, an R-ring (M, μ, η),

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such that M is a right H-comodule and η : R → M and μ : M ⊗R M → M are right H-comodule maps. Using the notations m → m[0] ⊗R m[1] and m → m[0] ⊗L m[1] for the HL - and HR -coactions, respectively, H-colinearity of η and μ means the identities, for all m, m ∈ M, 1M [0] ⊗ 1M [1] = 1M ⊗ 1H , R

R

1M[0] ⊗ 1M[1] = 1M ⊗ 1H , L

L

(mm )[0] ⊗ (mm )[1] = m[0] m[0] ⊗ m[1] m[1] R

R

(mm )[0] ⊗ (mm )[1] = L

m[0] m[0] ⊗ m[1] m[1] . L

Symmetrically, a left H-comodule algebra is a monoid in H M. The functors in Remark 4.6 induced by the antipode are checked to be strictly antimonoidal. Therefore, the opposite of a right H-comodule algebra, with coactions in Remark 4.6, is a left H-comodule algebra and vice versa. Thus, there are four different categories of modules of a comodule algebra of a Hopf algebroid. Deﬁnition 4.11. Let H be a Hopf algebroid and M be a right H-comodule algebra. The left and right M-modules in MH are called left–right and right–right relative Hopf modules, respectively. Their categories are denoted by M MH and MH M , respectively. Left and right M op -modules in H M are called right–left and left–left relative Hopf modules, respectively, and their categories are denoted by H MM and H M M, respectively. Explicitly, for example, a right–right (M, H)-relative Hopf module is a right module W for the R-ring M such that the action is a right H-comodule map W ⊗R M → W . Using index notations, with superscripts for the HR -coactions and subscripts for the HL -coactions, both on W and M, this amounts to the following identities, for w ∈ W and m ∈ M, (w · m)[0] ⊗ (w · m)[1] = w[0] · m[0] ⊗ w[1] m[1] and R

R

(w · m)[0] ⊗ (w · m)[1] = w[0] · m[0] ⊗ w[1] m[1] . L

L

In contrast to relative Hopf modules of bialgebroids in Section 3.7.1, relative Hopf modules of Hopf algebroids cannot be identiﬁed with comodules of a coring. Still they determine an adjunction, very similar to (3.16). Consider a right comodule algebra M of a Hopf algebroid H = (HL , HR , S) over base algebras L and R. Denote the HR -coinvariant subalgebra of M by N. It follows from Proposition 4.8 that for any right N-module V , V ⊗N M is a right–right relative Hopf module via the second factor. The resulting functor − ⊗N M : MN → MH M turns out to have a right adjoint. H R Any object W in MM can be regarded as an object in MH M , so we can take its HR coinvariants (cf. Section 4.3.2). These considerations lead to an adjoint pair of functors − ⊗ M : MN → MH M N

and (−)coHR : MH M → MN .

(4.6)

The unit of the adjunction is given, for any right N-module V , by the map V → (V ⊗ M)coHR , N

v → v ⊗ 1M N

(4.7)

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and the counit is given, for an (M, H)-relative Hopf module W , by W coHR ⊗ M → M, N

w ⊗ m → w · m. N

(4.8)

The message of this observation is that by studying descent theory of Galois extensions of a Hopf algebroid H = (HL , HR , S), one can examine the adjunction in both (4.6), corresponding to an H-comodule algebra M, and (3.16) determined by M regarded as an HR -comodule algebra. Proposition 4.12 is obtained by observing that the units of the two adjunctions coincide, and the counit of the adjunction in (4.8) is obtained by restricting to the objects in MH M the counit of the adjunction (3.16). Proposition 4.12. Consider a Hopf algebroid H = (HL , HR , S) and a right H-comodule algebra M. Denote the HR -coinvariant subalgebra of M by N. R (1) The functor − ⊗N M : MN → MH M is fully faithful if and only if the functor H − ⊗N M : MN → MM is fully faithful. R (2) If the functor −⊗N M : MN → MH M is an equivalence, then also the functor H − ⊗N M : MN → MM is an equivalence.

4.3.4. The fundamental theorem of Hopf modules In this section, we investigate the adjunction (4.6) in a special case. The coproducts L and R in a Hopf algebroid H make the underlying algebra H a right H-comodule algebra. The corresponding right–right relative Hopf modules are called simply Hopf modules, and their category is denoted by MH H . The coinvariants of the right HR -comodule algebra H are the elements tR (r), for r ∈ R, where tR is the target map. If HR is the underlying bialgebroid in a Hopf algebroid (HL , HR , S), then tR : Rop → H, equivalently, sL : L → H, is a right HR -Galois extension (cf. Section 4.6.2). Hence, Theorem 4.13, known as the Fundamental Theorem of Hopf modules, can be interpreted as a descent Theorem for this Galois extension L ∼ = Rop ⊆ H. Theorem 4.13. For a Hopf algebroid H over base algebras L and R, the functor − ⊗L H : ML → MH H is an equivalence. Theorem 4.13 is proved by constructing the inverses of the unit (4.7) and the counit (4.8) of the relevant adjunction. Use the notations for the structure maps of a Hopf algebroid in Deﬁnition 4.1. For a right L-module V , the inverse of (4.7) is the map (V ⊗L H)coHR → V , i vi ⊗L hi → i vi · sL (L (hi )). For a Hopf module W , denote the HR -coaction by w → w[0] ⊗R w[1] and for the HL -coaction, write w → w[0] ⊗L w[1] . Then, an epimorphism W → W coHR is given by w → w[0] · S(w[1] ). The inverse of (4.8) is the map W → W coHR ⊗L H, w → w[0] [0] · S(w[0] [1] ) ⊗L w[1] . 4.4. Integral theory In a Hopf algebra H over a commutative ring k, integrals are invariants of the regular module of the underlying k-algebra H, with respect to the character given by the

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215

counit. The study of integrals provides a lot of information about the structure of the k-algebra H. Most signiﬁcantly, (k-relative) semisimplicity of H is equivalent to its separability over k and also to the existence of a normalized integral in H [57,58]. Since this extends Maschke’s theorem about group algebras, it is known as a Maschke-type theorem. Its dual version relates cosemisimplicity and coseparability of the k-coalgebra underlying H to the existence of normalized cointegrals [57]. Another group of results concerns Frobenius property of H, which is equivalent to the existence of a nondegenerate integral in H [59, 60]. Integral theory of Hopf algebras was generalized to Hopf algebroids in [47]. Deﬁnition 4.14. For an algebra R, consider a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1. Right integrals in B are the invariants of the right regular module of the underlying R-ring (B, s), with respect to the right character . Equivalently, invariants of the right regular module of the Rop -ring (B, t), with respect to . That is, the elements of B = {i ∈ B | ib = is((b)),

∀b ∈ B} = {i ∈ B | ib = it((b)),

∀b ∈ B} ∼ = HomB (R, B). A right integral i is normalized if (i) = 1R . Symmetrically, left integrals in a left R-bialgebroid are deﬁned as invariants of the left regular module of the underlying R-ring or Rop -ring, with respect to the left character deﬁned by the counit. Note that a left integral i in a left bialgebroid B is a left integral also in Bcop and a right integral in the right bialgebroids B op op and Bcop . Since the counit in a right (resp. left) bialgebroid is a right (resp. left) character, there is no way to consider left (resp. right) integrals in a right (resp. left) bialgebroid. On the contrary, the base algebra in a right bialgebroid B is both a right and a left B-comodule via coactions given by the source and target maps, respectively. Hence, there are corresponding notions of left and right cointegrals. Deﬁnition 4.15. For an algebra R, consider a right R-bialgebroid B, with structure maps denoted as in Deﬁnition 3.1. A right cointegral on B is an element of HomB (B, R) = {ι ∈ HomR (B, R) | (ι ⊗ B) ◦ = s ◦ ι}. R

A right cointegral ι is normalized if ι(1B ) = 1R . Symmetrically, a left cointegral on B is an element of B Hom(B, R). Left and right cointegrals on a left R-bialgebroid B are deﬁned analogously as left and right comodule maps B → R. If a right R-bialgebroid B is ﬁnitely generated and projective as a, say right, R-module (via right multiplication by the source map), then the isomorphism HomR (R, B∗ ) ∼ = HomR (B, R) induces an isomorphism B∗ Hom(R, B∗ ) ∼ = B Hom (B, R). Hence in this case, left integrals in the left R-bialgebroid B ∗ are the

216

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same as right cointegrals on B. Similar statements hold for all other duals of left and right bialgebroids. For a Hopf algebroid (HL , HR , S), left (resp. right) cointegrals on HL and HR can be shown to be left (resp. right) H-comodule maps. 4.4.1. Maschke-type theorems Recall that an R-ring B is said to be separable provided that the multiplication map B ⊗R B → B is a split epimorphism of B-bimodules. The R-ring B is said to be left (resp. right) semisimple (or sometimes R-relatively semisimple) if every left (resp. right) B-module is R-relative projective. That is, every B-module epimorphism, which has an R-module section, is a split epimorphism of B-modules. By a classical result due to Hirata and Sugano [61], a separable R-ring is left and right semisimple. For Hopf algebroids also the converse can be proved. Theorem 4.16. Let (HL , HR , S) be a Hopf algebroid over base algebras L and R with structure maps as denoted in Deﬁnition 4.1. The following properties are equivalent. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii)

The R-ring (H, sR ) underlying HR is separable. The Rop -ring (H, tR ) underlying HR is separable. The L-ring (H, sL ) underlying HL is separable. The Lop -ring (H, tL ) underlying HL is separable. The R-ring (H, sR ) underlying HR is right semisimple. The Rop -ring (H, tR ) underlying HR is right semisimple. The L-ring (H, sL ) underlying HL is left semisimple. The Lop -ring (H, tL ) underlying HL is left semisimple. There exists a normalized right integral in HR . There exists a normalized left integral in HL . The counit R in HR is a split epimorphism of right H-modules. The counit L in HL is a split epimorphism of left H-modules.

Since the source map in a right R-bialgebroid is a right R-module section of the counit, implication (v)⇒(xi) is obvious. For a right H-module section ν of the counit, ν(1R ) is a normalized right integral. Thus, (xi)⇒(ix). The antipode in a Hopf algebroid maps a normalized right integral in HR to a normalized left integral in HL and vice versa. So (ix)⇔(x). If i is a normalized left integral in HL , then the map H → H ⊗R H, h → hi(1) ⊗R S(i(2) ) = i(1) ⊗R S(i(2) )h is an H-bimodule section of the multiplication in the R-ring underlying HR (where the index notation R (h) = h(1) ⊗R h(2) is used for h ∈ H). This proves (x)⇒(i). The remaining equivalences follow by symmetry. Note that equivalences (iv)⇔(viii)⇔(x) ⇔(xii) hold also for a ×L -Hopf algebra HL (as discussed in Section 4.6.2). As an alternative to Theorem 4.16, one can ask about properties of the R ⊗k Rop ring, underlying a right bialgebroid HR , and the L ⊗k Lop -ring, underlying a left bialgebroid HL , in a Hopf algebroid (HL , HR , S). Theorem 4.17 is obtained by application of [39, Theorem 6.5]. For a k-algebra L, consider a left L-bialgebroid HL . Denote its L ⊗k Lop -ring structure by (H, sL , tL ) and its L-coring structure by

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217

(H, L , L ). Look at L as a left L ⊗k Lop -module, with action given by left and right multiplications. Look at H as a right L ⊗k Lop -module, with action given by right multiplications by sL and tL . Note that H ⊗L⊗k Lop L ∼ = H/{ hsL (l) − htL (l) | h ∈ H, l ∈ L }

(4.9)

is an L-coring (via quotient maps of L and L ) and a left H-module. Hence, we can speak about the invariants of H ⊗L⊗k Lop L with respect to L . An invariant of H ⊗L⊗k Lop L is said to be normalized if the quotient of L maps it to 1L . Theorem 4.17. Consider a Hopf algebroid (HL , HR , S) over base k-algebras L and R. The following assertions are equivalent. (i) (ii) (iii) (iv) (v) (vi)

The R ⊗k Rop -ring underlying HR is separable. The L ⊗k Lop -ring underlying HL is separable. The R ⊗k Rop -ring underlying HR is right semisimple. The L ⊗k Lop -ring underlying HL is left semisimple. There is a normalized invariant in the right H-module R ⊗R⊗k Rop H. There is a normalized invariant in the left H-module H ⊗L⊗k Lop L.

For a ×L -Hopf algebra HL (as discussed in Section 4.6.2), equivalences (ii)⇔(iv)⇔(vi) in Theorem 4.17 also hold. Recall that an R-coring B is said to be coseparable provided that the comultiplication map B → B ⊗R B is a split monomorphism of B-bicomodules. The R-coring B is said to be left (resp. right) cosemisimple (or sometimes R-relatively cosemisimple) if every left (resp. right) B-comodule is R-relative injective. That is, every B-comodule monomorphism, which has an R-module retraction, is a split monomorphism of B-comodules. A coseparable R-coring is left and right cosemisimple. For Hopf algebroids also the converse can be proved. Theorem 4.18. For a Hopf algebroid (HL , HR , S) over base algebras L and R, the following properties are equivalent. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv)

The R-coring underlying HR is coseparable. The L-coring underlying HL is coseparable. The R-coring underlying HR is right cosemisimple. The R-coring underlying HR is left cosemisimple. The L-coring underlying HL is right cosemisimple. The L-coring underlying HL is left cosemisimple. There exists a normalized right cointegral on HR . There exists a normalized left cointegral on HR . There exists a normalized right cointegral on HL . There exists a normalized left cointegral on HL . The source map in HR is a split right HR -comodule monomorphism. The target map in HR is a split left HR -comodule monomorphism. The source map in HL is a split left HL -comodule monomorphism. The target map in HL is a split right HL -comodule monomorphism.

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4.4.2. Frobenius Hopf algebroids It was proven by Larson and Sweedler in [59] that every ﬁnite-dimensional Hopf algebra over a ﬁeld is a Frobenius algebra. Although this is not believed to be true for any ﬁnitely generated projective Hopf algebra over a commutative ring, Frobenius Hopf algebras form a distinguished class.AHopf algebra is known to be a Frobenius algebra if and only if it possesses a nondegenerate integral [60]. It is a self-dual property; a nondegenerate integral determines a nondegenerate cointegral, that is, a nondegenerate integral in the dual Hopf algebra. In a Hopf algebroid, there are four algebra extensions: the ones given by the source and target maps of the two constituent bialgebroids. Among Hopf algebroids, those in which these are Frobenius extensions play an even more distinguished role. Although the dual of any ﬁnitely generated projective Hopf algebroid is not known to be a Hopf algebroid, duals of Frobenius Hopf algebroids are Frobenius Hopf algebroids. While every ﬁnitely generated and projective Hopf algebra over a commutative ring k was proved by Pareigis to be a quasi-Frobenius k-algebra in [60], an analogous statement fails to hold for Hopf algebroids. In [47, Section 5], Hopf algebroids were constructed, which are ﬁnitely generated and projective over their base algebras (in all the four senses in (3.3) and (3.5)) but are not quasi-Frobenius extensions of the base algebra. Recall (e.g. from [62]) that an R-ring (H, s) is said to be Frobenius provided that H is a ﬁnitely generated and projective left R-module, and ∗ H := R Hom(H, R) is isomorphic to H as an H-R bimodule. Equivalently, (H, s) is a Frobenius Rring if and only if H is a ﬁnitely generated and projective right R-module and H ∗ := HomR (H, R) is isomorphic to H as an R-H bimodule. These properties are equivalent also to the existence of an R-bimodule map ψ : H → R, the so called e ⊗ Frobenius functional, possessing a dual basis i R fi ∈ H ⊗R H, satisfying, for i all h ∈ H, i ei · ψ(fi h) = h = i ψ(hei ) · fi . The following characterization of Frobenius Hopf algebroids was obtained in [47, Theorem 4.7], see the Corrigendum. Theorem 4.19. Consider a Hopf algebroid H over base algebras L and R with structure maps denoted as in Deﬁnition 4.1. Assume that H is a ﬁnitely generated and projective left and right R-module via the actions in (3.3) and a ﬁnitely generated and projective left and right L-module via the actions in (3.5). The following statements are equivalent. The R-ring (H, sR ) is Frobenius. The Rop -ring (Hop , tR ) is Frobenius. The L-ring (H, sL ) is Frobenius. The Lop -ring (Hop , tL ) is Frobenius. ι:H → There exists a right cointegral ι on HR such that the map HomR (H, R), h → ι(h−) is bijective. (vi) There exists a left cointegral υ on HL such that the map υ:H → Hom(H, L), h → υ(−h) is bijective. L i : L Hom(H, L) → H, (vii) There exists a right integral i in HR such that the map ψ → tL (ψ(i(2) ))i(1) is bijective. j : HomR (H, R) → H, (viii) There exists a left integral j in HL such that the map φ → j (2) tR (φ(j (1) )) is bijective. (i) (ii) (iii) (iv) (v)

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A Hopf algebroid for which these equivalent conditions hold is said to be Frobenius, and a right (resp. left) integral obeying property (vii) (resp. (viii)) is said to be nondegenerate. In a Frobenius Hopf algebroid the antipode S is bijective. j )−1 (1H ) is a right If j is a nondegenerate left integral in HL , then ι := ( cointegral on HR and a Frobenius functional for the R-ring (H, sR ) with dual basis j (1) ⊗R S(j (2) ). Thus, (viii)⇒(i). If property (i) holds, then the right cointegrals on HR are shown to form a free rank one left R-module I, via the action r · ι = ι(tR (r)−). Using ﬁnitely generated projectivity of the right R-module H, the dual H ∗ := HomR (H, R) can be equipped with a Hopf module structure, with coinvariants I. Hence, Theorem 4.13 implies an isomorphism H ∗ ∼ = H ⊗R I. This isomorphism is used to show that the cyclic generator ι of the R-module I satisﬁes condition (v). If there is a right cointegral ι as in part (v), then a nondegenerate left integral j as in part (viii) is constructed in terms of ( ι )−1 and a dual basis for the ﬁnitely ι ◦ S, which generated projective right R-module H. It is shown to satisfy ( j )−1 = implies bijectivity of S. The remaining equivalences follow by relations between the source and target maps in HL and HR and symmetrical versions of the arguments above. For a Frobenius Hopf algebroid (HL , HR , S) over base algebras L and R, all the four duals HomR (H, R), R Hom(H, R), HomL (H, L), and L Hom(H, L) possess j (left or right) bialgebroid structures. A left integral j in HL , such that the map in part (viii) of Theorem 4.19 is bijective, determines further similar isomorphisms R Hom(H, R) → H, Hom L (H, L) → H and L Hom(H, L) → H. What is more, putting ι := ( j )−1 (1H ), there is an algebra automorphism of H, ζ : H → H,

h → h(2) tR (ι(j h(1) )).

(4.10)

These isomorphisms combine to bialgebroid (anti-) isomorphisms between the four duals of H (cf. [46, Theorem 5.16]). The Frobenius property of a Hopf algebroid was shown to be self-dual in [46, Theorem 5.17 and Proposition 5.19] in the following sense. Theorem 4.20. Consider a Frobenius Hopf algebroid (HL , HR , S) over base algej in part (viii) of bras L and R. Let j be a left integral in HL such that the map Theorem 4.19 is bijective. Then, the left R-bialgebroid H ∗ := HomR (H, R) extends to a Hopf algebroid. A bijective antipode is given in terms of the map (4.10) by j )−1 ◦ S ◦ ζ ◦ j. The right bialgebroid structure is determined by the requireS ∗ := ( ment that S ∗ is a bialgebroid anti-isomorphism in the sense of Proposition 4.4. This dual Hopf algebroid is Frobenius, with nondegenerate left integral ( j )−1 (1H ) ∈ H∗ . In a paper [63] by Szlachányi, an equivalent description of a Frobenius Hopf algebroid (HL , HR , S) was proposed, via the so-called double algebras. In this picture, the isomorphism j in part (viii) of Theorem 4.19 is used to transfer the multiplication in H ∗ := HomR (H, R) to a second algebra structure in H, with unit j. In this way, four Frobenius ring structures are obtained on H. Note that the coproducts in HL and HR correspond canonically to the Frobenius ring structures transferred from H ∗ . In

220

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this approach, the Hopf algebroid axioms are formulated as compatibility conditions between the two algebra structures on H. 4.5. Galois theory of Hopf algebroids Galois extensions by Hopf algebras are the same as Galois extensions by the constituent bialgebra. Still, since the structure of a Hopf algebra is more complex than that of a bialgebra, it allows to derive stronger results. On the contrary, Galois theory of a Hopf algebroid is more conceptually different from the Galois theory of the constituent bialgebroids. As it is discussed in Section 4.3, comodules of a Hopf algebroid carry more structure than comodules of any constituent bialgebroid. Consequently, Galois theory of Hopf algebroids, discussed in this section, is signiﬁcantly richer than the theory of bialgebroids. In particular, for a comodule algebra of a Hopf algebroid (cf. Section 4.3.3), several theorems concerning an equivalence between the category of relative Hopf modules and the category of modules of the coinvariant subalgebra, that is descent theorems, can be proved. By Deﬁnition 4.10 and Theorem 4.9, a right comodule algebra M of a Hopf algebroid H = (HL , HR , S) is both an HL -comodule algebra and an HR -comodule algebra. Denote the HR -coinvariant subalgebra of M by N. In light of Proposition 4.8, there are two corresponding canonical maps M ⊗ M → M ⊗ H,

m ⊗ m → mm[0] ⊗ m[1]

M ⊗ M → M ⊗ H,

m ⊗ m → m[0] m ⊗ m[1] ,

N

N

R

L

N

N

R

L

and

(4.11) (4.12)

where m → m[0] ⊗R m[1] and m → m[0] ⊗L m[1] denote the HR -coaction and the HL -coaction on M, respectively. In general, the bijectivity of the two canonical maps (4.11) and (4.12) are not known to be equivalent. Only a partial result [54, Lemma 3.3] is known. Proposition 4.21. If the antipode S in a Hopf algebroid (HL , HR , S) is bijective, then the HR -canonical map (4.11) is bijective if and only if the HL -canonical map (4.12) is bijective. This follows by noting that the two canonical maps differ by the isomorphism A from (4.5). By Propositions 4.8 and 4.21, for a right comodule algebra M of a Hopf algebroid (HL , HR , S) with a bijective antipode, an algebra extension N ⊆ M is HL -Galois if and only if it is HR -Galois. Remark 4.22. Consider a Hopf algebroid H = (HL , HR , S) over base algebras L and R, which is ﬁnitely generated and projective as a left L-module via the source map in HL and as a left R-module via the target map in HR . Then, H is in particular ﬂat as a left L-module and as a left R-module. Therefore, by Theorem 4.7 and Theorem 4.9,

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221

the category of right H-comodules is strict monoidally isomorphic to the category of comodules for both the constituent left and right bialgebroids HL and HR . Thus, comodule algebras for H coincide with comodule algebras for HL or HR . Furthermore, if the antipode is bijective, it follows from Propositions 4.8 and 4.21 that an algebra extension N ⊆ M is a right HR -Galois extension if and only if it is a right HL -Galois extension. 4.5.1. Depth 2 Frobenius extensions An analog of Theorem 3.27 for Frobenius Hopf algebroids is [12, Theorem 3.6]. Theorem 4.23. An algebra extension N ⊆ M is a right Galois extension by a Frobenius Hopf algebroid (i.e. by any of its constituent bialgebroids) if and only if it is a Frobenius extension, it is balanced, and it satisﬁes the (left and right) depth 2 conditions. In a case of a Frobenius extension N ⊆ M, the left and right depth 2 properties are equivalent. By Remark 4.22, for a Frobenius Hopf algebroid (HL , HR , S), the HL - and HR -Galois properties of an extension are equivalent. By Theorem 3.24, a right depth 2 and balanced algebra extension N ⊆ M are Galois extension by a right bialgebroid (M ⊗N M)N . In addition, if N ⊆ M is a Frobenius extension, bialgebroid in a Frobenius Hopf then (M ⊗N M)N is shown to be a constituent right algebroid. A nondegenerate (left and right) integral i mi ⊗N mi ∈ (M ⊗N M)N is provided by the dual basis of a Frobenius functional ψ : M → N. A nondegenerate (left and right) integral in the dual Hopf algebroid N End N (M) is provided by ψ. In the converse direction, note that a right comodule M of a Hopf algebroid H is an H∗ -module. If H is a Frobenius Hopf algebroid and N ⊆ M is an H-Galois extension, a Frobenius functional M → N is given by the action of a nondegenerate integral in H∗ . 4.5.2. Cleft extensions by Hopf algebroids For an algebra M and a coalgebra C over a commutative ring k, Homk (C, M) is a k-algebra via the convolution product (f g)(c) := f(c(1) )g(c(2) ),

for f, g ∈ Homk (C, M), c ∈ C.

(4.13)

A comodule algebra M of a k-Hopf algebra H is said to be a cleft extension of its coinvariant subalgebra N provided that there exists a convolution invertible map j ∈ Homk (H, M), which is an H-comodule map. The relevance of cleft extensions by Hopf algebras stems from Doi and Takeuchi’s observation in [64] that N ⊆ M is a cleft extension if and only if it is a Galois extension and an additional normal basis property holds, that is, M ∼ = N ⊗k H as a left N-module right H-comodule. What is more (establishing an even stronger similarity with Galois extensions of ﬁelds), N ⊆ M is a cleft extension if and only if M is isomorphic to a crossed product of N with H with respect to an invertible 2-cocycle [64, 65]. The results have been extended to Hopf algebroids in [48]. To formulate the definition of a cleft extension, as a ﬁrst step, a generalized convolution product has to be introduced. Using notations as in Deﬁnition 4.1, in a Hopf algebroid (HL , HR , S)

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there is an L-coring (H, L , L ) and an R-coring (H, R , R ). Consider an L⊗k R-ring M, with multiplications μL : M ⊗L M → M and μR : M ⊗R M → M. For these data, the convolution algebra (4.13) can be generalized to a convolution category. It has two objects, conveniently labelled by L and R. For P, Q ∈ {L, R}, morphisms from P to Q are Q-P bimodule maps H → M, where the bimodule structure of the domain is determined by the (P- and Q-) coring structures of H and the bimodule structure of the codomain is determined by the (P- and Q-) ring structures of M. For P, Q, T ∈ {L, R}, and morphisms f : Q → P and g : T → Q, composition is given by a convolution product f g := μQ ◦ (f ⊗ g) ◦ Q . Q

(4.14)

Recall from Theorem 4.9 that a right comodule algebra of a Hopf algebroid (HL , HR , S) has a canonical R-ring structure over the base algebra R of HR . Deﬁnition 4.24. Consider a Hopf algebroid H = (HL , HR , S) over base algebras L and R. A right H-comodule algebra M is said to be a cleft extension of the HR coinvariant subalgebra N provided that the following properties hold. (i) The canonical R-ring structure of M extends to an L ⊗k R-ring structure. (ii) There exists an invertible morphism j : R → L in the convolution category (4.14), which is a right H-comodule map. In an H-cleft extension N ⊆ M, N can be proved to be an L-subring of M. In a Hopf algebroid H, using the notations introduced in Deﬁnition 4.1, an L ⊗k Rring is given by (H, sL , sR ). The identity map of H is a morphism R → L in the corresponding convolution category (4.14). It is obviously right H-colinear. What is more, the antipode is its inverse by axioms (iii) and (iv) in Deﬁnition 4.1. Hence, the right regular comodule algebra of a Hopf algebroid is a cleft extension of the coinvariant subalgebra tR (R). This extends a well-known fact that the right regular comodule algebra of a Hopf algebra, over a commutative ring k, is a cleft extension of k. A further similarity between cleft extensions by Hopf algebras and Hopf algebroids is expressed by the following Theorem 4.25. Theorem 4.25. Consider a Hopf algebroid (HL , HR , S) over base algebras L and R. Its right comodule algebra M is a cleft extension of the HR -coinvariant subalgebra N if and only if the following properties hold. (i) N ⊆ M is a Galois extension by HR ; (ii) the normal basis condition holds, that is, M ∼ = N ⊗L H as left N-modules right H-comodules. Note the appearance of the two base algebras L and R in conditions (i) and (ii) in Theorem 4.25. Another characterization of a cleft extension by a Hopf algebroid can be given by using the construction of a crossed product.

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Deﬁnition 4.26. Consider a left bialgebroid B over a base k-algebra L. Denote its structure maps as in Deﬁnition 3.3. We say that B measures an L-ring N with unit map ι : L → N if there exists a k-module map: ·B ⊗k N → N, the so-called measuring, such that, for b ∈ B, l ∈ L and n, n ∈ N, the following axioms are satisﬁed. (i) b · 1N = ι((b)), (ii) (t(l)b) · n = (b · n)ι(l) and (s(l)b) · n = ι(l)(b · n), (iii) b · (nn ) = (b(1) · n)(b(2) · n ). Note that in Deﬁnition 4.26, condition (iii) makes sense in view of (ii). Consider a left bialgebroid B over a k-algebra L and denote its structure maps as in Deﬁnition 3.3. Let N be an L-ring with unit ι : L → N, which is measured by B. These data determine a category C(B, N) as follows. Consider B ⊗k B as an L-bimodule via left multiplication by s and t in the ﬁrst factor. For an element f in L Hom L (B ⊗k B, N), consider the following (L-balancing) conditions. For a, b ∈ B, and l ∈ L, f(a ⊗ s(l)b) = f(as(l) ⊗ b) k k f(a ⊗ t(l)b) = f(at(l) ⊗ b) k k f(a ⊗ s(l)b) = (a(1) · ι(l))f(a(2) ⊗ b) k k f(a ⊗ t(l)b) = f(a(1) ⊗ b)(a(2) · ι(l)).

(T ◦) (S◦) (T •) (S•)

k

k

Deﬁne a category C(B, N) of two objects ◦ and •. For two objects X, Y ∈ {◦, •}, the morphisms X → Y are elements of L HomL (B ⊗k B, N), satisfying conditions (SX) and (TY ). The composition of the morphisms g : X → Y and f : Y → Z is given by (f g)(a ⊗ b) := f(a(1) ⊗ b(1) )g(a(2) ⊗ b(2) ). k

k

k

The unit morphism at the object ◦ is the map a ⊗k b → (ab) · 1N = ι((ab)) and the unit morphism at the object • is the map a ⊗k b → a · (b · 1N ). Deﬁnition 4.27. Consider a left bialgebroid B over a k-algebra L with structure maps as in Deﬁnition 3.3. Let N be a B-measured L-ring, with unit ι : L → N. An N-valued 2-cocycle on B is a morphism ◦ → • in the category C(B, N) given in the previous paragraph such that, for all a, b, c ∈ B, (i) σ(1B , b) = ι (b) = σ(b, 1B ), (ii) a(1) · σ(b(1) , c(1) ) σ(a(2) , b(2) c(2) ) = σ(a(1) , b(1) )σ(a(2) b(2) , c). The B-measured L-ring N is called a σ-twisted B-module if in addition, for n ∈ N and a, b ∈ B, (iii) 1B · n = n, (iv) a(1) · (b(1) · n) σ(a(2) , b(2) ) = σ(a(1) , b(1) )(a(2) b(2) · n). An N-valued 2-cocycle σ on B is said to be invertible if it is invertible as a morphism in C(B, N).

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Note that in Deﬁnition 4.27, conditions (ii) and (iv) make sense in view of the module map and balanced properties of σ. If an L-ring N is measured by a left L-bialgebroid B, then the Lop -ring N op is measured by the co-opposite left Lop bialgebroid Bcop . The inverse of an N-valued 2-cocycle σ on B turns out to be an N op -valued 2-cocycle on Bcop . For a left bialgebroid B over an algebra L, with structure maps denoted as in Definition 3.3, the base algebra L is measured by B via the left action b · l := (bs(l)). For this measuring, conditions (S◦) and (S•) are equivalent and also conditions (T ◦) and (T •) are equivalent. Consequently, an L-valued 2-cocycle on B in the sense of Deﬁnition 4.27 is equivalent to a cocycle as considered in Section 3.4.2. Extending the cocycle double twists of Section 3.4.2, one can consider more general deformations of a Hopf algebroid H (or a ×L -Hopf algebra B, discussed in Section 4.6.2) by an N-valued invertible 2-cocycle σ in Deﬁnition 4.27 (cf. [45, Appendix]). In that construction, the base algebra L of HL is replaced by an HL -measured L-ring N. In particular, the Connes and Moscovici’s bialgebroids in Section 3.4.6 arise in this way. A crossed product N#σ B of a left L-bialgebroid B with a σ-twisted B-module N, with respect to an N-valued 2-cocycle σ, is the the L-module tensor product N ⊗L B (where B is a left L-module via the source map s), with associative and unital multiplication )#b(3) b(2) , (n#b)(n #b ) = n(b(1) · n )σ(b(2) , b(1)

for n#b, n #b ∈ N ⊗ B. L

Equivalence classes of crossed products with a bialgebroid were classiﬁed in [48, Section 4]. Theorem 4.28. A right comodule algebra M of a Hopf algebroid (HL , HR , S) is a cleft extension of the HR -coinvariant subalgebra N if and only if M is isomorphic, as a left N-module and right H-comodule algebra, to a crossed product algebra N#σ HL , with respect to some invertible N-valued 2-cocycle σ on HL . 4.5.3. The structure of Galois extensions by Hopf algebroids In the theory of Galois extensions by Hopf algebras (with a bijective antipode), important tools are provided by theorems, which state that in appropriate situations, surjectivity of the canonical map implies its bijectivity, that is, the Galois property of an algebra extension N ⊆ M. There are two big groups of such theorems. In the ﬁrst group, a Hopf algebra H is assumed to be a ﬂat module over its commutative base ring k and its regular comodule algebra is assumed to be a projective H-comodule. These properties hold in particular if H is a ﬁnitely generated and projective k-module, in which case such a theorem was proven ﬁrst by Kreimer and Takeuchi [66]. In the second group of such results, due to Schneider, H is assumed to be a projective k-module and its comodule algebra M is assumed to be a k-relative injective H-comodule [67]. Analogous results for extensions by a Hopf algebroid H = (HL , HR , S) were obtained in the papers [12, 54, 55], respectively. A common philosophy behind such theorems originates from a paper [43] of Schauenburg (on the Hopf algebra case).

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The key idea is to investigate a lifting (4.16) of the canonical map (4.11) introduced for an H-comodule algebra M in the following. By a general result [68, Theorem 2.1] about Galois comodules, split surjectivity of the lifted canonical map (4.16) as a morphism of relative (M, HR )-Hopf modules implies the HR -Galois property, that is, bijectivity of (4.11), whenever (M ⊗T M)coHR = M ⊗T M coHR , where (−)coHR denotes the HR -coinvariants functor. For a Hopf algebra H over a commutative ring k and a right H-comodule algebra M with coinvariant subalgebra N, the canonical map M ⊗N M → M ⊗k H can be lifted to a map M⊗M k

/ / M⊗M

can

N

/ M⊗H . k

(4.15)

More generally, consider a Hopf algebroid H = (HL , HR , S) over base k-algebras L and R. For a right H-comodule algebra M, the canonical map (4.11) can be lifted to M ⊗ M → M ⊗ H, T

R

m ⊗ m → mm[0] ⊗ m[1] T

R

(4.16)

for any k-algebra T such that the HR -coinvariant subalgebra N of M is a T-ring. The map (4.16) is a morphism of right–right (M, H)-relative Hopf modules. Theorem 4.29. Consider a Hopf algebroid H over base k-algebras L and R, with a bijective antipode. Denote its structure maps as in Deﬁnition 4.1. Assume that H is a ﬂat left R-module (via right multiplication by tR ) and a projective right HR -comodule (via R ). Let M be a right H-comodule algebra with HR -coinvariant subalgebra N. Under these assumptions the following statements hold. (1) If the HR -coinvariants of the right H-comodule M ⊗k M (with coactions given via the second factor) are precisely the elements of M ⊗k N, then the canonical map (4.11) is bijective if and only if it is surjective. (2) If the canonical map (4.11) is bijective, then M is a projective right N-module. Since coinvariants are deﬁned as a kernel, the coinvariants of M ⊗k M are precisely the elements of M ⊗k N if, for example, M is a ﬂat k-module. To gain an impression of the proof of part (1) of Theorem 4.29, note that ﬂatness of the left R-module H and projectivity of the right regular HR -comodule together imply that M ⊗R H is projective as a right–right (M, HR )-relative Hopf module. Hence, if the canonical map (4.11) is surjective, then the (surjective) lifted canonical map (4.16) is a split epimorphism of right–right (M, HR )-relative Hopf modules, for any possible k-algebra T. Thus, bijectivity of the canonical map (4.11) follows by [68, Theorem 2.1]. Part (2) of Theorem 4.29 follows by exactness of the naturally equivalent functors HomHR (H, −) ∼ = HomN (M, (−)coHR ), which is a consequence of the projectivity of the right regular HR -comodule. If in a Hopf algebroid H with a bijective antipode, H is a ﬁnitely generated and projective left and right R-module, and a ﬁnitely generated and projective left and right L-module, with respect to the respective actions in (3.3) and (3.5),

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then it is obviously a ﬂat left R-module. Furthermore, in this case, the right regular HR -comodule can be shown to be projective by using Theorem 4.13. The following Weak Structure Theorem 4.30 is, thus, based on Remark 4.22, part (2) of Theorem 4.29 and its application to the right comodule algebra M op of the opposite Hopf algebroid Hop , and a theorem [69, Theorem 3.5] about Galois corings. Theorem 4.30. Consider a Hopf algebroid H = (HL , HR , S) over base algebras L and R, with a bijective antipode S. Assume that H is a ﬁnitely generated and projective left and right R-module, and a ﬁnitely generated and projective left and right L-module, with respect to the respective actions in (3.3) and (3.5). For a right H-comodule algebra M, with HR -coinvariant subalgebra N, the following statements are equivalent. (i) (ii) (iii) (iv) (v) (vi)

The extension N ⊆ M is HR -Galois. ∼ HR ∼ HL M is a generator in the category MH M = MM = MM . H The HR -coinvariants functor MM → MN is fully faithful. The extension N ⊆ M is HL -Galois. M is a generator in the category M MH ∼ = M MHR ∼ = M MHL . H The HL -coinvariants functor M M → N M is fully faithful.

Furthermore, if these equivalent conditions hold, then M is a projective left and right N-module. It was a key observation by Doi that relative injectivity of a comodule algebra M of a Hopf algebra H is equivalent to the existence of the so-called total integral – meaning an H-comodule map j : H → M such that j(1H ) = 1M [70]. This fact extends to Hopf algebroids. Recall (e.g. from [71]) that, for any functor U : A → B, between any categories A and B, an object A ∈ A is said to be U-injective if the map HomA (g, A) : HomA (Y, A) → HomA (X, A) is surjective, for any objects X, Y ∈ A, and all morphisms g ∈ HomA (X, Y) such that U(g) is a split monomorphism in B. If U has a right adjoint, then U-injectivity of an object A is equivalent to the unit of the adjunction, evaluated at A, being a split monomorphism in A (see [71, Proposition 1]). For example, for a Hopf algebra H over a commutative ring k, injective objects with respect to the forgetful functor MH → Mk are precisely relative injective H-comodules. A version of Theorem 4.31 below was proved in [55, Theorem 4.1], using the notion of relative separability of a forgetful functor. Recall from Remark 4.6 that the opposite of a right comodule algebra M of a Hopf algebroid H has a canonical structure of a left H-comodule algebra. Theorem 4.31. For a right comodule algebra M of a Hopf algebroid H = (HL , HR , S), the following statements are equivalent. (i) There exists a right H-comodule map (resp. right HL -comodule map) j : H → M, such that j(1H ) = 1M .

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(ii) M is injective with respect to the forgetful functor MH → ML (resp. with respect to the forgetful functor MHL → ML , i.e. M is a relative injective HL -comodule). (iii) Any object in the category MH M of right–right relative Hopf modules is injective with respect to the forgetful functor MH → ML (resp. with respect to the forgetful functor MHL → ML ). If in addition the antipode S is bijective, then assertions (i)–(iii) are also equivalent to (iv) There exists a left H-comodule map (resp. left HR -comodule map) j : H → M such that j (1H ) = 1M . Hence, assertions (i)–(iv) are also equivalent to the symmetrical versions of (ii) and (iii). The key idea behind Theorem 4.31 is the observation that both forgetful functors MH → ML and MHL → ML possess left adjoints − ⊗L H (cf. (4.3)). A correspondence can be established between comodule maps j as in part (i) and natural retractions of the counit of the adjunction, that is, of the HL -coaction. Based on [72, Theorem 4.7] and Theorem 4.31, also the following Strong Structure Theorem holds. Theorem 4.32. Consider a Hopf algebroid (HL , HR , S) over base algebras L and R, with a bijective antipode S. Assume that H is a ﬁnitely generated and projective left and right R and L-module with respect to the actions in (3.3) and (3.5). For a right HR - (equivalently, right HL -), Galois extension N ⊆ M, the following statements are equivalent. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)

M is a faithfully ﬂat right N-module. The inclusion N → M splits in MN . M is a generator of right N-modules. M is a faithfully ﬂat left N-module. The inclusion N → M splits in N M. M is a generator of left N-modules. The functor − ⊗N M : MN → MH M is an equivalence. H M is a projective generator in MM . There exists a right H-comodule map j : M → H such that j(1H ) = 1M .

Note that if a Hopf algebra H over a commutative ring k is a projective k-module, then M ⊗k H is a projective left M-module for any right H-comodule algebra M. Thus, denoting the subalgebra of coinvariants in M by N, surjectivity of the canonical map M ⊗N M → M ⊗k H implies that its lifted version (4.15) is a split epimorphism of left M-modules, so in particular of k-modules. By Schneider’s result [67, Theorem I], if the antipode of H is bijective and M is a k-relative injective right H-comodule, then bijectivity of the canonical map follows from the k-module splitting of its lifted version (4.15). To formulate the following generalization, Theorem 4.33, of this

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result, note that the lifted canonical map (4.16) is an L-bimodule map, with respect to the L-actions l·(m⊗T m )·l := m·R (sL (l))⊗T R (tL (l ))·m (recall that the R-, and T-actions on M commute by virtue of (3.14)) and l·(m⊗R h)·l := m⊗R sL (l)tL (l )h, on its domain and codomain, respectively, where the notations from Deﬁnition 4.1 are used. Theorem 4.33. Consider a Hopf algebroid H with a bijective antipode over base algebras L and R. Denote its structure maps as in Deﬁnition 4.1. Let M be a right H-comodule algebra with HR -coinvariants N. Let T be a k-algebra such that N is a T-ring. In this setting, if the lifted canonical map (4.16) is a split epimorphism of right L-modules, then the following statements are equivalent. (i) N ⊆ M is an HR -Galois extension and the inclusion N → M splits in MN . (ii) N ⊆ M is an HR -Galois extension and the inclusion N → M splits in N M. (iii) There exists a right H-comodule map j : H → M such that j(1H ) = 1M . (iv) M ⊗N − : N M → in MN .

HM M

is an equivalence and the inclusion N → M splits

Furthermore, if the equivalent properties (i)–(iii) hold, then M is a T-relative projective right N-module. Note that by Propositions 4.8 and 4.21, in parts (i) and (ii) the HR -Galois property can be replaced equivalently by the HL -Galois property. Also, by Theorem 4.31, the existence of a unit preserving right H-comodule map in part (iii) can be replaced equivalently by the existence of a unit preserving left H-comodule map. The most interesting part of Theorem 4.33 is perhaps the claim that if property (iii) holds, then right L-module splitting of the lifted canonical map (4.16) implies the HR -Galois property. The proof of this fact is based on an observation, originally in [55], that assertion (iii) is equivalent to relative separability of the forgetful funcH tor MH → ML , with respect to the forgetful functor MH M → M . A relative separable functor reﬂects split epimorphisms in the sense that if f is a morphism in MH M , which is a split epimorphism of right L-modules, then it is a split epimorphism of right H-comodules. This proves that, under the assumptions made, the lifted canonical map (4.16) is a split epimorphism of right H-comodules. Furthermore, the H forgetful functor MH M → M possesses a left adjoint − ⊗R M. Hence, the rightright (M, H)-relative Hopf module M ⊗R H, which is isomorphic to H ⊗R M by bijectivity of the antipode, is relative projective in the sense that a split epimorphism g in MH of codomain H ⊗R M ∼ = M ⊗R H is a split epimorphism in MH M . This proves that in the situation considered the lifted canonical map (4.16) is a split epimorphism of right-right (M, H)-relative Hopf modules. Then it is a split epimorphism of right-right (M, HR )-relative Hopf modules. Moreover, in terms of a unit preserving right H-comodule map H → M, a left N-module splitting of the equalizer of the HR -coaction on M and the map m → m ⊗R 1H can be constructed. This implies that the equalizer is preserved by the functor M ⊗T −, that is, (M ⊗T M)coHR = M ⊗T N. Thus, the canonical map (4.11) is bijective by [68, Theorem 2.1].

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Recall that a left module V of a k-algebra N is k-relative projective if and only if the left action N ⊗k V → V is a split epimorphism of left N-modules. If V has an additional structure of a right comodule for a k-coalgebra C such that the N-action is a right C-comodule map, then it can be asked if the action N ⊗k V → V splits as a map of left N-modules and right C-comodules as well. In the case when it does, V is said to be a C-equivariantly projective left N-module. For a Galois extension N ⊆ M by a k-Hopf algebra H, H-equivariant projectivity of the left N-module M was shown by Hajac to be equivalent to the existence of a strong connection [73]. That is, interpreting a Hopf Galois extension as a noncommutative principal bundle, equivariant projectivity means its local triviality. In case of a Galois extension N ⊆ M by a k-Hopf algebra H, the equivalent conditions (i)–(iii) in Theorem 4.33 are known to imply H-equivariant projectivity of the left N-module M. To obtain an analogous result for a Galois extension by a Hopf algebroid, slightly stronger assumptions are needed, see Theorem 4.35 below. Deﬁnition 4.34. Consider a Hopf algebroid H and consider a T-ring N for some algebra T. Let V be a left N-module and right H-comodule such that the left N-action on V is a right H-comodule map. V is said to be a T-relative H-equivariantly projective left N-module provided that the left action N ⊗T V → V is an epimorphism split by a left N-module, right H-comodule map. Theorem 4.35. Consider a Hopf algebroid H with a bijective antipode. Let M be a right H-comodule algebra with HR -coinvariants N. Let T be an algebra such that N is a T-ring. Assume that there exists a unit preserving right H-comodule map H → M and that the lifted canonical map (4.16) is a split epimorphism of L-bimodules. Then N ⊆ M is a right HR -, and right HL -Galois extension and M is a T-relative Hequivariantly projective left N-module. Under the assumptions of Theorem 4.35, the HR -Galois property holds by Theorem 4.33 and the HL -Galois property follows by Propositions 4.8 and 4.21. A proof of Theorem 4.35 is completed by constructing the required left N-module right H-comodule section of the action N ⊗T M → M. The construction makes use of the relative Hopf module section of the lifted canonical map (4.16). The existence of such a section was proved in the paragraph following Theorem 4.33. Examples of L-relative H-equivariantly projective Galois extensions are provided by cleft extensions by a Hopf algebroid H with a bijective antipode over base algebras L and R.

4.6. Alternative notions In the literature there is an agreement that the right generalization of a bialgebra to the case of a noncommutative base algebra is a bialgebroid. On the contrary, there is some discussion about the structure that should replace a Hopf algebra. In this current ﬁnal section we revisit and compare the various suggestions.

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4.6.1. Lu Hopf algebroid In Deﬁnition 4.1, the antipode axioms are formulated for a compatible pair of a left and a right bialgebroid. In following Deﬁnition 4.36, quoted from [6], only a left bialgebroid is used. While the ﬁrst one of the antipode axioms in Deﬁnition 4.1 (iv) is easily formulated also in this case, to formulate the second one, some additional assumption is needed. Deﬁnition 4.36. Consider a left bialgebroid B, over a k-algebra L, with structure maps denoted as in Deﬁnition 3.3. B is a Lu Hopf algebroid provided that there exists an antialgebra map S : B → B and a k-module section ξ of the canonical epimorphism B ⊗k B → B ⊗L B such that the following axioms are satisﬁed. (i) S ◦ t = s, (ii) μB ◦ (S ⊗L B) ◦ = t ◦ ◦ S, (iii) μB ◦ (B ⊗k S) ◦ ξ ◦ = s ◦ , where μB denotes multiplication in the L-ring (B, s) and μB denotes multiplication in the underlying k-algebra B. None of the notions of a Hopf algebroid in Deﬁnition 4.1 or 4.36 seem to be more general than the others. Indeed, a Hopf algebroid in the sense of Deﬁnition 4.1, which does not satisfy the axioms in Deﬁnition 4.36, is constructed as follows. Let k be a commutative ring in which 2 is invertible. For the order 2 cyclic group Z2 , consider the group bialgebra kZ2 as a left bialgebroid over k. Equip it with the twisted (bijective) antipode S, mapping the order 2 generator t of Z2 to S(t) := −t. Together with the unique right bialgebroid, determined by the requirement that S is a bialgebroid anti-isomorphism in the sense of Proposition 4.4, they constitute a Hopf algebroid as in Deﬁnition 4.1. However, for this Hopf algebroid there exists no section ξ as in Deﬁnition 4.36. 4.6.2. On ×L -Hopf algebras The coinvariants of the (left or right) regular comodule of a bialgebra H over a commutative ring k are precisely the multiples of the unit element 1H . H is known to be a Hopf algebra if and only if H is an H-Galois extension of k. Indeed, the hom-tensor relation H HomH (H ⊗k H, H ⊗k H) ∼ = Homk (H, H) relates the inverse of the canonical map to the antipode. Motivated by this characterization of a Hopf algebra, in [33] Schauenburg proposed the following deﬁnition. Deﬁnition 4.37. Let B be a left bialgebroid over an algebra L, with structure maps denoted as in Deﬁnition 3.3. Consider the left regular B-comodule, whose coinvariant subalgebra is t(Lop ). B is said to be a ×L -Hopf algebra provided that the algebra extension t : Lop → B is left B-Galois. The notion of a ×L -Hopf algebra in Deﬁnition 4.37 is more general than that of a Hopf algebroid as in Deﬁnition 4.1. Indeed, consider a Hopf algebroid (HL , HR , S), over the base algebras L and R, with structure maps denoted as in Deﬁnition 4.1. The canonical map H ⊗Lop H → H ⊗L H, h ⊗ h → h(1) ⊗ h(2) h is bijective, with inverse h ⊗ h → h(1) ⊗ S(h(2) )h . Hence HL is a ×L -Hopf algebra. An example

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of a ×L -Hopf algebra, that admits no Hopf algebroid structure, was proposed in [76, Example 8]. Extending a result [74] by Schauenburg about Hopf algebras, the following proposition was proved in [44]. Proposition 4.38. Consider a left bialgebroid B over a base algebra L. If there is a left B-Galois extension N ⊆ M such that M is a faithfully ﬂat left L-module, then B is a ×L -Hopf algebra. Indeed, the canonical maps can : M ⊗N M → B⊗L M and ϑ : B⊗Lop B → B⊗L B satisfy the pentagonal identity (B ⊗L can) ◦ (can ⊗N M) = (ϑ ⊗L M) ◦ can13 ◦ (M ⊗N can), where the (well deﬁned) map can13 : M ⊗N (B ⊗L M) → (B ⊗Lop B) ⊗L M is obtained by applying ‘can’ to the ﬁrst and third factors. In Theorem 3.13, bialgebroids were characterized via strict monoidality of a forgetful functor. A characterization of a similar ﬂavor for ×L -Hopf algebras was given in [33, Theorem and Deﬁnition 3.5]. Recall that a monoidal category (M, ⊗, U) is said to be right closed if the endofunctor − ⊗ X on M possesses a right adjoint denoted by hom(X, −) for any object X in M. The monoidal category of bimodules of an algebra L is right closed with hom(X, Y ) = HomL (X, Y ). It is slightly more involved to see that so is the category of left B-modules, for a left L-bialgebroid B, with hom(X, Y ) = B Hom(B ⊗L X, Y ) (where for a left B-module X, B ⊗L X is a left B-module via the diagonal action and a right B-module via the ﬁrst factor). A strict monoidal functor F : M → M between right-closed categories is called strong right closed provided that the canonical morphism F(hom(X, Y )) → hom(F(X), F(Y )) is an isomorphism, for all objects X, Y ∈ M. Theorem 4.39. A left bialgebroid B over a base algebra L is a ×L -Hopf algebra if and only if the (strict monoidal) forgetful functor B M → L ML is strong right closed. For a Hopf algebra H over a commutative ring k, those (left or right) H-modules, which are ﬁnitely generated and projective k-modules, possess (right or left) duals in the monoidal category of (left or right) H-modules. This property extends to ×L -Hopf algebras (hence to Hopf algebroids!) as follows. Proposition 4.40. Let B be a ×L -Hopf algebra over a base algebra L. Denote its structure maps as in Deﬁnition 3.3. For a left B-module M, the dual M ∗ := HomL (M, L) is a left B-module via the action (b · φ)(m) := b1 t(φ(b2 · m)) , for b ∈ B, φ ∈ M ∗ , m ∈ M, where for the inverse of the (right B-linear) canonical map B ⊗Lop B → B ⊗L B, the index notation b ⊗ b → b1 ⊗ b2 b is used. Furthermore, if M is a ﬁnitely generated and projective right L-module, then M ∗ is a right dual of M in the monoidal category B M.

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4.6.3. Hopf monad In Theorem 3.5, bialgebroids were related to bimonads on a bimodule category that possess a right adjoint. In the paper [39], special bimonads, the so-called Hopf monads, on autonomous categories were studied. Recall that a monoidal category is said to be left (resp. right) autonomous provided that every object possesses a left (resp. right) dual. In particular, a category of ﬁnitely generated and projective bimodules is autonomous. Instead of the somewhat technical deﬁnition in [39, 3.3], we adopt an equivalent description from [39, Theorem 3.8] as a deﬁnition. Deﬁnition 4.41. A left (resp. right) Hopf monad is a bimonad B on a left (resp. right) autonomous monoidal category M such that the left (resp. right) autonomous structure of M lifts to the category of B-algebras. The reader should be warned that although the same term Hopf monad is used in the papers [39] and [26], they have different meanings (and a further totally different meaning of the same term is used in [20]). Also, the notions of a comodule and a corresponding (co)integral in [39] are different from the notions used in these notes. 4.6.4. A ∗ -autonomous structure on a strong monoidal special opmorphism between pseudomonoids in a monoidal bicategory In the paper [28], strong monoidal special opmorphisms h in monoidal bicategories, from a canonical pseudomonoid Rop ⊗ R to some pseudomonoid B, were studied. The opmorphism h was called Hopf if in addition there is a ∗ -autonomous structure on B and h is strong ∗ -autonomous (where Rop ⊗ R is meant to be ∗ -autonomous in a canonical way). In [28, Section 3], a bialgebroid was described as a strong monoidal special opmorphism h of pseudomonoids in the monoidal bicategory of [Algebras; Bimodules; Bimodule maps]. This opmorphism h is strong ∗ -autonomous if and only if the corresponding bialgebroid constitutes a Hopf algebroid with a bijective antipode (see [46, Section 4.2]). Acknowledgment The author’s work is supported by the Bolyai János Fellowship and the Hungarian Scientiﬁc Research Fund OTKA F67910. She is grateful for their helpful comments to Dénes Bajnok, Imre Bálint, Tomasz Brzezin´ ski, and Kornél Szlachányi. References [1] J. Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. 121 (2) (1985) 1–39. [2] J. Mrˇcun, The Hopf algebroids of functions on étale groupoids and their principal Morita equivalence, J. Pure Appl. Algebra 160 (2001) 249–262. [3] J. Donin,A. Mudrov, Quantum groupoids and dynamical categories, J.Algebra 296 (2006) 348–384. [4] P. Etingof, D. Nikshych, Dynamical quantum groups at roots of 1, Duke Math J. 108 (2001) 135–168. [5] G. Karaali, On Hopf Algebras and their Generalizations, preprint arXiv:math.QA/0703441.

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Comodules and Corings Tomasz Brzezin´ ski Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K. E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Categorical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The category of comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Special types of corings and comodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Coseparable and cosplit corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Frobenius corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Galois corings (comodules) and generalized descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Morita theory and corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Bialgebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Hopf modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Noncommutative differential geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Noncommutative algebraic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Extensions and dualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Weak and lax corings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. C-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Other topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00206-4 237

238 239 244 244 246 252 254 254 261 263 268 268 271 275 281 286 288 288 293 297 301 301 304 309 311 311

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1. Introduction A coring is one of the most basic algebraic structures (formally) dual to that of a ring (but also a ring itself can be viewed as a coring). The notion of a coring appeared ﬁrst in the algebra literature in 1975 in Sweedler’s paper [178] on a predual version of the Jacobson–Bourbaki correspondence. As corings are coalgebras in the category of bimodules of an algebra, they were implicitly studied earlier, for example, in [115]. Also, in the late 1970s, they appeared under the name BOCS in representation theory and were used in studies of matrix problems in works of Drozd, Kleiner, and Roiter [161]. Between the late 1970s and the end of the 1990s corings attracted rather limited attention, with, to the best of our knowledge, only a handful of papers [106, 122, 134, 135, 183, 184] and a monograph [20] published. In the meantime, there was a resurgence of interest in Hopf algebras, triggered by the appearance of quantum groups as symmetries of integrable systems in quantum and statistical mechanics in works of Drinfeld and Jimbo. This led to intensive studies of Hopf algebras also from a purely algebraic point of view and to the development of more general categories of Hopf-type modules. These serve as representations of Hopf algebras and related structures, such as those described by solutions to braid or Yang-Baxter equations. By the end of the 1990s, all such Hopf-type modules were uniﬁed in terms of entwined modules associated to entwining structures (cf. [36,49]). It was an observation of Takeuchi (reported in [136]) that entwined modules are comodules of a certain coring that instigated renewal of interest in corings in Hopf algebra theory. This allowed one to view more conceptually and directly a connection between properties of various types of Hopf modules [37]. More or less at the same time corings appeared as one of the main structures in an approach to noncommutative algebraic geometry advocated by Kontsevich and Rosenberg [124]. The aim of this chapter is two-fold. First, we would like to give a summary of basic deﬁnitions and basic properties of corings and comodules, focussing on recent developments. Rather than giving a complete picture, we give an overview of the coring and comodule theory, describe basic examples, and discuss applications which prompted recent resurgence of interest in this theory. Since a coalgebra is an example of a coring, a summary of properties of comodules of a coring provides also the reader with the knowledge of properties of comodules of a coalgebra. Furthermore, many types of comodules of coalgebras considered in Hopf algebra theory (Hopf-type modules) turn out to be comodules of speciﬁc corings. In this way, this chapter should supplement [140], which is our second aim. For fuller and more detailed description of corings and comodules, the reader should consult [53]. Inevitably, there is an overlap between this chapter and [53], in particular, in these parts which deal with a pure or “older” coring theory. On the other hand, we have tried to include as much as possible results, which have been obtained or about which we have learnt since [53] was written. The reader should be aware that since coring theory went through a signiﬁcant transformation over the last few years and is still being intensively developed, this chapter is a “state-of-theart” presentation of these developments rather than a deﬁnitive account. It is still

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too early to determine the signiﬁcance of particular developments. The selection of topics reﬂects the author’s taste more than their actual “objective” value (whatever this might be). 2. Categorical preliminaries Throughout this chapter we use the language of category theory. On the other hand, to understand the main elements of the theory developed in this chapter, very little knowledge of categorical language is needed. The aim of this section is to introduce the reader who is not familiar with this language to the basic concepts and notions. We hope that in this way such a reader can freely read the later parts of this chapter. The reader familiar with category theory can skip this section, and only come back to it when notation or conventions used later on will raise doubts. A standard reference for category theory is [132]. 2.1. Categories. A category A consists of a class of objects, and, for any two objects A, B, a set of morphisms or arrows from A to B, denoted by A(A, B). The sets of arrows satisfy the following axioms: (a) A(A, B) ∩ A(C, D) = ∅, if either A = C or B = D. (b) For any objects A, B, C and any morphisms f ∈ A(A, B) and g ∈ A(B, C), there exists a unique morphism g ◦ f ∈ A(A, C) known as a composition of f with g and such that for any object D and any h ∈ A(C, D), (h ◦ g) ◦ f = h ◦ (g ◦ f ). (c) For any object A, there is a unique identity morphism id A ∈ A(A, A) such that, for any object B and any f ∈ A(A, B), g ∈ A(B, A), id A ◦ g = g, f ◦ id A = f. In this chapter, we adopt the convention whereby for any object A the identity morphism id A is denoted by A. For any f ∈ A(A, B), we write f : A → B. A composition is also denoted as a sequence of arrows A → B → C. The opposite category Aop of A is a category with the same collection of objects and arrows as A, but any arrow A → B in A is understood as an arrow B → A, that is, Aop (A, B) = A(B, A). Given two categories A and B, the product category A×B has ordered pairs (A, B), with A an object in A and B an object in B, as objects and with morphism sets: A × B((A, B), (A , B )) = A(A, A ) × B(B, B ). A subcategory B of A is a category whose objects are a subclass of objects of A, and, for each pair of objects in B, the morphism set is a subset of the morphism set of corresponding objects in A. A subcategory B of A is said to be full if B(A, B) = A(A, B), for all objects A, B of B. If k is a commutative ring with unit, by a k-linear category, we understand a category A in which all morphism sets are k-modules and composition of morphisms is k-linear. In case k = Z, a k-linear category is called a preadditive category. All the categories described in the main body of this chapter are k-linear.

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2.2. Examples of categories. Here are some elementary examples of categories: (1) Set (the category of sets): objects are sets and morphisms are functions. (2) R M, MR , R MS (the category of left, resp. right, R-modules, over a ring R, resp. (R, S )-bimodules over rings R and S): objects are left, resp. right, R-modules, resp. (R, S )-bimodules, morphisms are R-module, resp. (R, S )bimodule maps. Morphisms in these categories are denoted by R Hom(−, −), resp. HomR (−, −), and R HomS (−, −). (3) Any monoid G can be understood as a category G with a single object ∗. The set of morphisms G(∗, ∗) equals G; the composition of morphisms is simply the multiplication in G. Similarly, any associative, unital k-algebra can be understood as a k-linear category with one object. 2.3. Types of morphisms. A morphism f : A → B in A is called (a) a monomorphism if, for all g, h ∈ A(C, A), f ◦ g = f ◦ h implies g = h; (b) an epimorphism if, for all g, h ∈ A(B, D), g ◦ f = h ◦ f implies g = h; (c) a retraction if there exists g ∈ A(B, A) with f ◦ g = B; a retraction is necessarily an epimorphism and hence is also called a split epimorphism; (d) a section if there exists g ∈ A(B, A) with g ◦ f = A; a section is necessarily a monomorphism and hence is also called a split monomorphism; (e) an isomorphism if f is both a retraction and a section. The splitting morphism g as in (c) and (d) is called the inverse of f and denoted by f −1 . 2.4. Functors. A covariant functor or simply a functor F from A to B is a map assigning to each object A of A an object F(A) of B and to any arrow f ∈ A(A, B ) an arrow F(f ) ∈ B(F(A), F(B)) such that (a) for any object A of A, the morphism F(id A ) is the identity morphism for F(A); (b) for any f ∈ A(A, B), g ∈ A(B, C), F(g ◦ f ) = F(g) ◦ F(f ). A contravariant functor F from A to B means a functor from Aop to B. A functor F : A → B between k-linear categories is assumed to be a k-linear operation in the sense that for any two objects A and B in A, A(A, B) → B(F(A), F(B)) is a homomorphism of k-modules. In case k = Z, a k-linear functor is called an additive functor. Main examples of functors include the identity functor A : A → A, which sends objects A to A and morphisms f to f , and two hom-functors, deﬁned for any object A of A as follows. The covariant hom-functor A(A, −) : A → Set,

B → A(A, B),

A(A, f ) : u → f ◦ u,

and the contravariant hom-functor A(−, A) : Aop → Set,

B → A(B, A),

A(f, A) : u → u ◦ f.

2.5. Full and faithful functors. A functor is said to be faithful if it is injective when restricted to each set of morphisms and is said to be full if it is surjective when

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restricted to each set of morphisms. A full and faithful functor is referred to as a fully faithful functor. 2.6. Tensor (induction) functors. Given rings R and S, and an (R, S )-bimodule M, one can deﬁne tensor functors − ⊗R M : MR → MS and M ⊗S − : S M → R M. On objects N ∈ MR . N ⊗R M is a right S-module through the right multiplication in M, (n ⊗R m)s = n ⊗R ms. For any morphism f : N → N in MR , f ⊗R M is a morphism in MS by n ⊗R m → f(n) ⊗R m. The other tensor functor is deﬁned symmetrically. 2.7. Forgetful functors. A forgetful functor is a functor that “forgets” part of the structure of objects. For example, since every right R-module is a set, there is a functor from MR → Set, which views modules as sets and module maps as functions. If A is a k-algebra, then there is a forgetful functor from the category MA of right A-modules to the category of k-modules, which views objects in MA purely as k-modules. 2.8. Natural transformations. Given functors F, G : A → B, a natural transformation ϕ : F → G is a collection of morphisms in B, ϕA : F(A) → G(A) labeled by all objects A ∈ A and such that for any morphism f : A → B, ϕB ◦ F(f ) = G(f ) ◦ ϕA . In case all the ϕA are isomorphisms, one calls ϕ a natural isomorphism. Functors from A to B and natural transformations form a category denoted by Fun(A, B). 2.9. Adjoint functors. A pair of functors L : A → B, R : B → A is said to be an adjoint pair provided that there exists a pair of natural transformations, η : id A → RL and ψ : LR → id B such that ψL(A) ◦ L(ηA ) = L(A),

R(ψB ) ◦ ηR(B) = R(B).

L is called a left adjoint of R and R is called a right adjoint of L. The transformation η is called a unit of the adjunction and ψ is termed a counit of the adjunction. L, R is a pair of adjoint functors if and only if there is a natural isomorphism : B(L(−), −) → A(−, R(−)) of functors Aop × B → Set. In terms of the unit and counit of the adjunction, is given by A,B = B(L(A), B) → A(A, R(B)),

g → R(g) ◦ ηA ,

and the inverse of is −1 A,B : A(A, R(B)) → B(L(A), B), f → ψB ◦ L(f ). In an adjoint pair (L, R), the functor L sends epimorphisms to epimorphisms (i.e. L preserves epimorphisms), while R sends monomorphisms to monomorphisms (i.e. R preserves monomorphisms). Furthermore, L is fully faithful if and only if the unit of the adjunction is a natural isomorphism, while R is fully faithful if and only if the counit is an isomorphism.

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The hom-tensor adjunction is one of the key adjoint pairs of functors in this chapter. Take R, S to be rings and M to be an (R, S )-bimodule. Then, the tensor functor − ⊗R M : MR → MS (cf. 2.6) is the left adjoint of the homfunctor HomS (M, −) : MS → MR . Here, HomS (M, V ) is a right R-module by (fr)(m) = f(rm), for all f ∈ HomS (M, V ), m ∈ M, and r ∈ R. The counit and unit of this adjunction are ψV : HomS (M, V) ⊗R M → V, ηX : X → HomS (M, X ⊗R M),

f ⊗R m → f(m), x → [m → x ⊗R m].

The isomorphisms X,V : HomS (X ⊗R M, V ) → HomR (X, HomS (M, V )) come out as X,V (g)(x) : m → g(x ⊗R m), for all right R-modules X, right S-modules V , g ∈ HomS (X ⊗R M, V ), x ∈ X, and m ∈ M. 2.10. Equivalence of categories. An adjoint pair of functors L : A → B, R : B → A with counit and unit, which are natural isomorphisms, is called a pair of inverse equivalences. Each of L and R is called an equivalence, and the categories A and B are said to be equivalent. 2.11. Comonads and monads. Let A be a category. A functor G : A → A is called a comonad if there exist a natural transformation δ : G → GG and a natural transformation σ : G → id A such that for all objects A ∈ A, the following diagrams δA

G(A)

δA / GG(A) G(A) R RRR RRG(A) RRR G(σA ) δA RRR R( σG(A) / G(A) GG(A)

/ GG(A) G(δA )

δA

GG(A)

/ GGG(A) ,

δG(A)

are commutative. A coalgebra or comodule of a comonad (G, δ, σ) is a pair (A, A ), where A is an object in A and A is a morphism A : A → G(A) rendering commutative the following diagrams A A

G(A)

A

/ G(A) δA

/ GG(A) ,

G(A )

A

/ G(A) A E EE EE σA E A EE " A.

A morphism of G-coalgebras (A, A ), (B, B ) is an element f ∈ A(A, B) such that B ◦ f = G(f ) ◦ A . G-coalgebras with their morphisms form a category denoted by AG .

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Dual to comonads one deﬁnes monads as functors F : A → A together with natural transformations μ : FF → F and η : id A → F such that for all objects A ∈ A, the following diagrams FFF(A)

F(μA )

μF(A)

FF(A)

/ FF(A) μA

μA

/ A,

η

F(A) / FF(A) F(A) Q QQQ QQF(A) QQQ μA F(ηA ) QQQ ( μA / F(A) FF(A)

are commutative. To monads one associates the category of algebras or modules, denoted by AF .1 In this chapter, we are primarily interested in comonads and coalgebras. 2.12. Comonads and adjoint functors. Any adjoint pair of functors L : A → B, R : B → A gives rise to a comonad (G, δ, σ) on B, where G = LR, and for all objects B in B, δB = L(ηR(B) ), σB = ψB . Here, η is the unit and ψ is the counit for the adjunction (L, R). Conversely, any comonad (G, δ, σ) on A gives rise to a pair of adjoint functors L : AG → A, R : A → AG . The functor L is the forgetful functor, while R is deﬁned by R(A) = (G(A), δA ). The unit and counit of adjunction are η(A,A ) = A and ψB = σB , for every object B of A and a G-coalgebra (A, A ). Dually, a pair of adjoint functors L : A → B, R : B → A gives rise to a monad F = RL : A → A. Conversely, a monad F induces a pair of adjoint functors A → AF , AF → A. 2.13. Monoidal categories and monoidal functors. A category A is called a monoidal category if there exist a functor − − : A × A → A, a distinguished object E in A and natural isomorphisms α : (− −) − → − (− −),

λ : E − → id A ,

: − E → id A

such that for all objects W, X, Y, Z in A, the following two diagrams commute: ((W X) Y ) Z

αW,X,Y Z

αW X,Y,Z

(W X) (Y Z)

αW,X,Y Z

/ (W (X Y )) Z SSSS SSαSW,X Y,Z SSSS SSSS ) W ((X Y ) Z) k kkkk k k k kkk ukkkk W αX,Y,Z / W (X (Y Z)) ,

1 The convention to denote coalgebras (comodules) of a comonad G by A while algebras of a monad G by AF is a standard category theory convention, which, unfortunately, is opposite to the module-theoretic conventions. We hope this will not cause any confusion later on.

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X,E,Y / X (E Y ) (X E) N Y NNN pp NNN ppp p N p p X Y NNN & xppp X λY X Y.

A monoidal category is denoted by (A, , E). The functor is referred to as a multiplication and E is called a unit. For example, if R is a ring, then the category of R-bimodules is monoidal with the usual tensor product of R-bimodules ⊗R as a multiplication and unit object R. A monoidal functor between categories (A, , E) and (A , , E ) is a triple (F, F 2 , F 0 ), where F : A → A is a functor, F 2 : F(−) F(−) → F(− −) is a natural transformation, and F 0 : E → F(E) is a morphism in A . F 2 and F 0 are compatible with the structure maps α, λ, and α , λ , in an obvious (natural) way; see [132, Section 11.2] for more details. (F, F 2 , F 0 ) is said to be strong if F 2 and F 0 are isomorphisms and strict if they are identities.

3. Corings From now on, A is an associative unital algebra over an associative commutative ring k with unit. The product in A is denoted by μA and the unit, understood both as a map k → A or as an element of A, by 1A . Unless explicitly speciﬁed otherwise, whenever we say “an algebra” we mean an associative k-algebra with unit. The unadorned tensor product is over k, that is, ⊗ = ⊗k . 3.1. Deﬁnition In this section, we give the deﬁnition and categorical interpretation of corings. 3.1. Deﬁnition of a coring [178]. An A-coring is an A-bimodule C with A-bilinear maps C : C → C ⊗A C

and

εC : C → A,

called coproduct and counit, rendering commutative the following diagrams C

C

C

C ⊗A C

C ⊗A C

/ C ⊗A C

C⊗A C

/ C ⊗A C ⊗A C ,

C

/ C ⊗A C C QQQ QQQ QQQC QQQ C εC ⊗A C QQQ Q( C⊗A εC / C. C ⊗A C

The ﬁrst of these diagrams is usually referred to as the coassociative law or it is said that it expresses the coassociativity of the coproduct. The diagrams involving εC express the counitality of the coproduct. In these diagrams, the obvious isomorphisms A ⊗A C C C ⊗A A are suppressed.

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3.2. Coassociativity and the Sweedler–Heyneman notation. The Sweedler– Heyneman notation allows one to express the action of the coproduct on elements. Given c ∈ C, one writes C (c) = c(1) ⊗A c(2) (with an implicit summation; the summation sign reminds about it). The coassociative law means that c(1) ⊗A c(2)(1) ⊗A c(2)(2) = c(1)(1) ⊗A c(1)(2) ⊗A c(2) , and therefore one can relabel the Sweedler indices and write c(1) ⊗A c(2) ⊗A c(3) . (C ⊗A C ) ◦ C (c) = For a detailed discussion of the Sweedler–Heyneman notation, we refer to [140, pp. 610–611]. 3.3. Morphisms of corings. Let C be an A-coring with coproduct C and counit εC , and let D be an A-coring with coproduct D and counit εD . An A-bimodule map f : C → D is said to be a morphism of A-corings or an A-coring morphism provided D ◦ f = (f ⊗A f ) ◦ C ,

εD ◦ f = εC .

In Sweedler’s notation, one thus requires that for all c ∈ C, f(c)(1) ⊗A f(c)(2) = f(c(1) ) ⊗A f(c(2) ),

εD (f(c)) = εC (c).

3.4. Coideals. The kernel of a surjective A-coring map p : C → D is called a coideal in C. Thus, an A-subbimodule K ⊂ C is a coideal if and only if D = C/K is a coring and p : C → C/K is a coring map. Equivalently, C (K) ⊆ ker(p ⊗A p) and εC (K) = 0. If the inclusion K ⊂ C is left and right C-pure (i.e. if it is a monomorphism after tensoring with C over A), the former condition is equivalent to C (K) = C ⊗A K + K ⊗A C. 3.5. Corings and comonads. For an A-bimodule C, the following statements are equivalent: (a) C is an A-coring with comultiplication C and counit εC . (b) The functor − ⊗A C : MA → MA is a comonad with comultiplication δ such that δA = C and counit σ such that σA = εC . (c) The functor C ⊗A − : A M → A M is a comonad with comultiplication δ such that δA = C and counit σ such that σA = εC . Sketch of proof. If C is a comultiplication for a coring C, the comultiplication δ for − ⊗A C (resp. C ⊗A −) is given by δM = M ⊗A C (resp. δM = C ⊗A M). If εC is the counit for C, then the counit σ for the comonad − ⊗A C (resp. C ⊗A −) is σM = M ⊗A εC (resp. σM = εC ⊗A M).

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3.6. The base algebra extension coring. Let D be a B-coring and let j : B → A be an algebra map. Thus A is a B-bimodule by bab = j(b)aj(b ). Then, C = A⊗B D⊗B A is an A-coring with the coproduct C : A ⊗B D ⊗B A → C ⊗A C A ⊗B D ⊗B A ⊗B D ⊗B A, a ⊗ B d ⊗B a → a ⊗B d (1) ⊗B 1A ⊗B d (2) ⊗B a , and counit εC : A ⊗B D ⊗B A → A, a ⊗B d ⊗B a → aj(εD (d ))a . 3.7. The category of corings. Corings over different rings can be grouped in the category of corings Crg. Objects in Crg are pairs (C, A), where A is a k-algebra and C is an A-coring. A morphism between corings (C, A) and (D, B) (cf. [101]) is a pair of maps (γ, α) satisfying the following conditions: (1) α : A → B is an algebra map. Thus one can view D as an A-bimodule via α. Explicitly, ada = α(a)dα(a ) for all a, a ∈ A and d ∈ D. (2) γ : C → D is an A-bimodule map such that the induced map, B ⊗A C ⊗A B → D,

b ⊗A c ⊗A b → bγ(c)b ,

is a morphism of B-corings, where B ⊗A C ⊗A B is the base algebra extension of C; see 3.6. 3.2. Examples In this section, we list a number of examples of corings, in particular those which appeared in the literature in recent years and which are related to other ﬁelds of algebra. 3.8. Coalgebras. If A is a commutative algebra and C is an A-module (i.e. both the left and right A-multiplications coincide, i.e. ac = ca, for all a ∈ A and c ∈ C), then an A-coring C is called a coalgebra. Thus any k-coalgebra is a k-coring (but not every k-coring is a k-coalgebra). Whatever can be said about a coring, can also be said about a k-coalgebra. 3.9. The trivial A-coring. An algebra A is itself an A-coring with both the coproduct and counit given by the identity map A. 3.10. The Sweedler coring [178]. Given an algebra map j : B → A, one deﬁnes the canonical Sweedler coring as follows. First, one views A as a B-bimodule via the algebra map j, i.e. bab = j(b)aj(b ), for all a ∈ A and b, b ∈ B. Then, one forms the A-bimodule C = A ⊗B A, with the A-multiplications a(a ⊗B a )a = aa ⊗B a a . The coproduct is deﬁned by C : C → C ⊗A C A ⊗B A ⊗B A,

a ⊗B a → a ⊗B 1A ⊗B a ,

and the counit is εC : C → A, a ⊗B a → aa . Even if both A and B are commutative algebras, the associated Sweedler coring need not be an A-coalgebra. The Sweedler coring can be understood as a base algebra extension coring 3.6, where D = B is a trivial coring.

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3.11. Comatrix corings [92]. Given algebras B and A, let P be a (B, A)-bimodule that is ﬁnitely generated and projective as a right A-module. Let ei ∈ P and πi ∈ P ∗ = HomA (P, A) be a ﬁnite dual basis for P. Write e = i ei ⊗A πi for the dual basis (canonical) element. The dual module P ∗ is an (A, B)-bimodule with multiplications (afb)(x) = af(bx), for all a ∈ A, b ∈ B, x ∈ P and f ∈ P ∗ . Thus C = P ∗ ⊗B P is an A-bimodule, a(f ⊗B x)a = af ⊗B xa . Similarly, P ⊗A P ∗ is a B-bimodule, and for all b ∈ B, be = eb. C is an A-coring with the coproduct and counit, C : P ∗ ⊗B P → P ∗ ⊗B P ⊗A P ∗ ⊗B P,

x ⊗B f → x ⊗B e ⊗B f,

and εC : P ∗ ⊗B P → A, f ⊗B x → f(x). The coring C is called a (ﬁnite) comatrix coring associated to a (B, A)-bimodule P. One way of understanding a comatrix coring is to observe that since P is a ﬁnitely generated projective right A-module, for all right A-modules N, HomA (P, N) N ⊗A P ∗ . Hence − ⊗B P : MB → MA is a left adjoint of − ⊗A P ∗ : MA → MB (this is the hom-tensor adjunction 2.9), with the unit given by the dual basis element e and the counit given by the evaluation map. Thus − ⊗A P ∗ ⊗B P : MA → MA is a comonad (see 2.12), and therefore, by 3.5, P ∗ ⊗B P is an A-coring with the structure given above. In case of an algebra map B → A, A is a (B, A)-bimodule that is (obviously) ﬁnitely generated and projective as an A-module. The dual module A∗ can be naturally identiﬁed with A. In this way, one can understand the Sweedler coring as a (ﬁnite) comatrix coring associated to A. 3.12. Inﬁnite comatrix corings I [93]. Let A be a (unital) k-algebra, and let P be a set ofﬁnitely generated and projective right A-modules. For any P ∈ P, write P ⊗ π P for the dual basis element of P. Set M = e eP = A i i P∈P P and let i {uP : M → M | P ∈ P} be the corresponding set of orthogonal idempotents. For a subalgebra T of the endomorphism algebra End A (M ), deﬁne B = P,Q∈P uP TuQ . Then B is a ﬁrm (nonunital) algebra and M is a (B, A)-bimodule. Deﬁne an (A, B)bimodule M† = HomA (P, A). P∈P

C=

M†

⊗B M is an A-coring with the coproduct and counit determined by C (f ⊗B x) = f ⊗B eP ⊗B x,

εC (f ⊗B x) = f(x),

for all f ∈ HomA (P, A) and x ∈ P. A construction of C from colimits is described in [58]. If P is a one-element or ﬁnite set, then C is a ﬁnite comatrix coring as deﬁned in 3.11. 3.13. Comatrix coring contexts [45, 103]. Let A be a unital algebra, and let B be a nonunital algebra that is ﬁrm2 or regular in the sense that the multiplication map 2 The terminologies ﬁrm ring and ﬁrm module are taken from [149].

248

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B ⊗B B → B is an isomorphism (of left or right B-modules). A comatrix coring context consists of (i) an (A, B)-bimodule N, which is ﬁrm as a right B-module in the sense that the product map N ⊗B B → N is an isomorphism of right B-modules; (ii) a (B, A)-bimodule M, which is ﬁrm as a left B-module in the sense that the product map B ⊗B M → M is an isomorphism of left B-modules; (iii) a pair of A- resp. B-bimodule maps σ : N ⊗B M → A,

τ : B → M ⊗A N,

such that the following diagrams N ⊗B MO ⊗A N

σ⊗A N

/ A ⊗A N

M ⊗A N ⊗B M o

N⊗B τ

N ⊗B B

/N

τ⊗B M

M⊗A σ

,

M ⊗A A

B ⊗B M

/M

commute. A comatrix coring context is denoted by (A, B, N, M, σ, τ). By [189, Theorem 2.51], the existence of a comatrix coring context (A, B, N, M, σ, τ) is equivalent to the statement that − ⊗B M : MB → MA and − ⊗A N : MA → MB is an adjoint pair of functors between the categories of ﬁrm right modules. In case B is a unital algebra (and then, by convention, M, N are unital B-modules), a comatrix coring context exists if and only if M is ﬁnitely generated and projective as a right A-module. The module N can be identiﬁed with the dual module M ∗ . With this identiﬁcation, σ is the evaluation map and τ is the map b → be, where e ∈ M ⊗A M ∗ is the dual basis element for M. 3.14. Corings from comatrix contexts [45,103]. Let (A, B, N, M, σ, τ) be a comatrix coring context. Write ∇N : N → N ⊗B B for the inverse of the product map N ⊗B B → N and ∇M : M → B ⊗B M for the inverse of the product map B ⊗B M → M. Then the A-bimodule C = N ⊗B M is an A-coring with the coproduct C = (N ⊗B τ ⊗B M) ◦ (∇N ⊗B M) = (N ⊗B τ ⊗B M) ◦ (N ⊗B ∇M ) and counit εC = σ. This is a consequence of the fact that − ⊗B M : MB → MA and − ⊗A N : MA → MB are an adjoint pair (compare the discussion in 3.11). In case of a unital ring B, the coring C constructed here is isomorphic to a (ﬁnite) comatrix coring 3.11. 3.15. Inﬁnite comatrix corings II [103]. Let A and B be (unital) k-algebras. Consider an (A, B)-bimodule N and a (B, A)-bimodule M. Let σ : N ⊗B M → A be an A-bimodule map so that M ⊗A N can be made into a nonunital k-algebra with product M ⊗A σ ⊗A N : M ⊗A N ⊗B M ⊗A N → M ⊗A N. Let R be a ﬁrm ring and suppose that there exists a (nonunital) algebra map τ : R → M ⊗A N. Then M is a left R-module with multiplication rm = (M ⊗A σ)(τ(r) ⊗B m) and N is a right R-module with multiplication nr = (σ ⊗A N )(n ⊗B τ(r)). If M is

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249

a ﬁrm left R-module (with the isomorphism denoted by ∇M : M → R ⊗R M), then C = N ⊗R M is an A-coring with the counit εC (n ⊗R m) = σ(n ⊗B m) and the coproduct C : N⊗R M → N⊗R M⊗A N⊗R M,

C = (N⊗R τ⊗R M )◦(N⊗R ∇M ).

The coring C is known as an inﬁnite comatrix coring. C is isomorphic to the coring constructed from the comatrix context (A, R, N ⊗R R, M, σ, τ). In the setup of 3.12, an inﬁnite comatrix coring is constructed by taking N = HomA (M, A); see [103, Section 6] for a detailed discussion of the relationship of inﬁnite comatrix corings to the corings described in 3.12. 3.16. Corings and semifree differential graded algebras [162]. A differential ∞ n (non-negatively) graded algebra is an N ∪ {0}-graded k-algebra = n=0 • •+1 together with a k-linear degree one operation d : → such that d ◦ d = 0 and, for all elements ω and all degree n elements ω, d(ωω ) = d(ω)ω +(−1)n ωd(ω ). The degree zero part 0 is a subalgebra of ; denote it by A. 1 is an A-bimodule. is said to be semifree, provided n = n−1 ⊗A 1 , for all n = 1, 2, . . .. Starting with a semifree differential graded algebra (A = 0 ), deﬁne C = Ag ⊕ 1 , where g is an indeterminate. In other words C is a direct sum of A and 1 as a left A-module. A compatible right A-module structure on C is deﬁned by (ag + ω)a = aa g + ada + ωa , for all a, a ∈ A, ω ∈ 1 . C is an A-coring with the coproduct speciﬁed by C (ag) = ag ⊗A g,

C (ω) = g ⊗A ω + ω ⊗A g − d(ω),

and the counit εC (ag + ω) = a, for all a ∈ A and ω ∈ 1 . 3.17. Frobenius extensions. The studies of Frobenius extensions of noncommutative rings were initiated in [119]. According to [120, 143], an algebra map A → B is called a Frobenius extension (of the ﬁrst kind) if and only if B is a ﬁnitely generated projective right A-module and B HomA (B, A) as (A, B)-bimodules. The (A, B)-bimodule structure of HomA (B, A) is given by (afb)(b ) = af(bb ), for all a ∈ A, b, b ∈ B and f ∈ HomA (B, A). The statement that A → B is a Frobenius extension is equivalent to the existence of an A-bimodule map E : B → A and an element β = i bi ⊗ b¯ i ∈ B ⊗A B such that, for all b ∈ B, E(bbi )b¯ i = bi E(b¯ i b) = b. i

i

E is called a Frobenius homomorphism and β is known as a Frobenius element. One can easily show that β ∈ (B ⊗A B)B := {x ∈ B ⊗A B | for all b ∈ B, bx = xb}. 3.18. Frobenius extensions and corings [116, Proposition 4.6]. An algebra map A → B is a Frobenius extension if and only if B is an A-coring such that the coproduct is a B-bimodule map.

250

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Sketch of proof. If A → B is a Frobenius extension with a Frobenius homomorphism E and a Frobenius element β, then B is an A-coring with the counit E and the B-bimodule coproduct B : B → B ⊗A B, b → bβ = βb. Conversely, if B is an A-coring with a B-bimodule coproduct B , then β = B (1B ) is a Frobenius element and the counit of B is a Frobenius homomorphism. This description of Frobenius extensions as corings generalizes the description of Frobenius algebras in terms of coalgebras [1]. The fact that a Frobenius extension induces a coring has been proved already in [191, Proposition 1.8.3]. 3.19. Depth-2 extensions [118]. An extension B ⊆ R of k-algebras is called a depth-2 extension if R ⊗B R is a direct summand in a ﬁnite direct sum of copies of R both as an (R, B)-bimodule and as a (B, R) bimodule. By [118, Lemma 3.7], this is equivalent to the existence of right and left D2 quasi-bases. A right D2 quasi basis consists of ﬁnite sets {γj }j∈J in B End B (R) and { m∈Mj r j m ⊗B r j m }j∈J in (R ⊗B R)B = {x ∈ R ⊗B R | ∀b ∈ B, bx = xb} such that rγj (r )cj m ⊗R cj m = r ⊗B r , j∈J,m∈Mj

for all r ⊗B r ∈ R ⊗B R. A left D2 quasi-basis is deﬁned symmetrically. Given a depth-2 extension B ⊆ R, deﬁne A = {a ∈ R | ∀b ∈ B, ab = ba}. Then C = B End B (R) is an A-coring with A-multiplications aca : r → ac(r)a , counit εC (c) = c(1R ), and coproduct γj ⊗A r j m c(r j m −). C (c) = j∈J

m∈Mj

Furthermore, D = (R ⊗B R)B is an A-coring with coproduct and counit obtained by restricting the structure maps of the Sweedler coring R ⊗B R. 3.20. Entwining structures [49]. A (right–right) entwining structure (over k) (A, C)ψ consists of a k-algebra (A, μA , 1A ), a k-coalgebra (C, C , εC ), and a k-module map ψ : C ⊗ A → A ⊗ C, rendering commutative the following bow-tie diagram: C ⊗ A ⊗MA C⊗ C⊗ @@ A MMM ~ qq8 @@ ~ q C⊗μ ⊗A A C M q ~ MMM @@ qqq ~~ M q @@ ~ M q ~ q & ψ⊗A ~ @@C⊗ψ ~ C ⊗ A @@ 8 M ~~ M q @@ ~ M MMεMC ⊗A ~ C⊗1Aqqqq @@ ~ M q ~ M q @@ ~ MMM q q q ~~ M qq & ψ A⊗C⊗ A C ⊗ A⊗C C A 8 M M q @@ MMM qqq @@ ~~ MMM q ~ q @@ M ~~ qqq 1A ⊗C MMM @@ ~~ & qqq A⊗εC @@ ~ ~ A @ 8 ⊗ CMMM ~~ ψ⊗C A⊗ψ @@ q ~ q @@ MMM qq ~~ @@ M qqμqA ⊗C ~~ q A⊗C MMM @ ~ q q & ~ A⊗C⊗C A⊗A⊗C

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251

The map ψ is known as an entwining map. C and A are said to be entwined by ψ. An entwining structure is a special case of a mixed distributive law between a comonad and a monad; see [18]. The comonad is − ⊗ C and the monad is − ⊗ A. 3.21. Corings associated to entwining structures [37, Proposition 2.2]. Let A be a k-algebra and C be a k-coalgebra with coproduct C and counit εC . For an entwining structure (A, C)ψ , view A ⊗ C as an (A, A)-bimodule with the left action a(a ⊗ c) = aa ⊗ c and the right action (a ⊗ c)a = a ψ(c ⊗ a), for all a, a ∈ A, c ∈ C. Then C = A ⊗ C is an A-coring with the coproduct C : C → C ⊗A C A ⊗ C ⊗ C,

C = A ⊗ C ,

and the counit εC = A ⊗ εC . Conversely, if A ⊗ C is an A-coring with the coproduct, counit, and left A-module structure above, then (A, C)ψ is an entwining structure, where ψ : c ⊗ a → (1 ⊗ c)a. 3.22. Weak entwining structures [56]. A (right–right) weak entwining structure (over k) is a triple (A, C, ψ) consisting of a k-algebra (A, μA , 1A ), a k-coalgebra (C, C , εC ), and a k-module map ψ : C ⊗ A → A ⊗ C, rendering commutative the following bow-tie diagram: C⊗ A ⊗OA C LL LLL OOO r8 ooo r o r C⊗1 C⊗1 ⊗A o OOψ⊗A A A LLL rr oo OOO r o L r o L r o OO' L r o w o & r C⊗μA / C ⊗ A C ⊗A/ C ⊗ C ⊗ A C⊗C C⊗A⊗A A⊗C⊗A

ψ⊗A

A⊗C⊗A

C⊗1A ⊗C

A⊗ψ

C ⊗ A ⊗OC

C⊗A⊗C

ψ

μA ⊗C

A⊗C

A⊗C⊗C

A⊗εC ⊗A

ψ⊗C

/ A⊗C / A⊗C⊗C A ⊗ A LLL rr8 ooo L r o r L o LL rrr oooμA A⊗εC LLL rrr A⊗εC ⊗C L& woooo

A⊗A⊗C

OOO OOO O ψ⊗C OOO '

C⊗ψ

A

.

One easily checks that any entwining structure is a weak entwining structure. 3.23. Corings associated to weak entwining structures [37, Proposition 2.3]. Let A be an algebra and C be a coalgebra with coproduct C and counit εC . Given a weak entwining structure (A, C, ψ), deﬁne p : A ⊗ C → A ⊗ C, and C = Im p =

i

p = (μA ⊗ C) ◦ (A ⊗ ψ) ◦ (A ⊗ C ⊗ 1A ),

ai ψ(ci ⊗ 1A ) |

i

ai ⊗ ci ∈ A ⊗ C .

C is an A-bimodule with the A-multiplications a i ai ⊗ ci a = i a ai ψ(ci ⊗ a ). Furthermore, C is an A-coring with coproduct C = (A ⊗ C )|C and counit εC = (A ⊗ εC )|C .

252

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3.3. Duality The notion of an A-coring is dual to that of an A-ring or an algebra over an algebra or an algebra extension. In this section, we explain this duality. 3.24. Rings over algebras. Given an algebra A, an A-ring is an A-bimodule B together with A-bimodule maps μB : B ⊗A B → B and ιB : A → B rendering commutative the following diagrams: B ⊗ A B ⊗A B

μB ⊗A B

μB

B⊗A μB

B ⊗A B

/ B ⊗A B

μB

/ B,

B⊗A ιB

/ B ⊗A B B RRR RRR RRRB μB RRR ιB ⊗A B RRR RR( μB / B B ⊗A B

The map μB is called a product and ιB is called a unit. The product map μB composed with the canonical surjection B ⊗ B → B ⊗A B induces a k-algebra structure on B. The unit for the k-algebra B is ιB (1A ). Thus the notion of an A-ring is equivalent to the existence of an algebra map ιB : A → B. 3.25. Dual rings [178, Proposition 3.2]. Let C be an A-coring. (1) C ∗ = HomA (C, A) is an algebra with unit εC and the product f ⊗A C g C f ∗r g : C −→ C ⊗A C −→ C −→ A, f ∗r g(c) = g(f(c(1) )c(2) ), and there is an antialgebra map ιR : A → C ∗ , a → εC (a−). Thus C ∗ is an Aop -ring. ∗ (2) C = A Hom(C, A) is an algebra with unit εC and the product f C⊗A g C f(c(1) g(c(2) )), f ∗l g : C −→ C ⊗A C −→ C −→ A, f ∗l g(c) = and there is an antialgebra map ιL : A → ∗ C, a → εC (−a). Thus ∗ C is an Aop -ring. (3) ∗ C ∗ = A HomA (C, A) is an algebra with unit εC and the product f ∗ g(c) = f(c(1) )g(c(2) ), and there is a ring morphism Z(A) → Z(∗ C ∗ ), a → εC (a−) = εC (−a) , where Z(A) is the center of A and Z(∗ C ∗ ) is the center of ∗ C ∗ . Thus ∗ C ∗ is a Z(A)-algebra. 3.26. Dual rings as natural endomorphism rings. (1) The algebra C ∗ in 3.25 is isomorphic to the algebra of natural endomorphisms of the forgetful functor A (−) : C M → A M, with the product given by ϕϕ = ϕ ◦ ϕ . (2) The algebra ∗ C in 3.25 is isomorphic to the algebra of natural endomorphisms of the forgetful functor (−)A : MC → MA .

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3.27. Monoidal functors and B|A-corings [184]. Corings and dual rings are closely related to monoidal functors between bimodule categories; see 2.13 for the deﬁnition of a monoidal category and a monoidal functor. Let A and B be k-algebras. Following [184, Deﬁnition 3.5], a B|A-coring is a pair (C, ι), where C is an A-coring and ι : B → ∗ C ∗ = A HomA (C, A) is a morphism of algebras. Here ∗ C ∗ is an algebra as in 3.25(3). As explained in [184, Theorem 3.6], there is a one-to-one correspondence between B|A-corings and representable monoidal functors between categories of bimodules (A MA , ⊗A , A) → (B MB , ⊗B , B), that is, monoidal functors (F, F 2 , F 0 ) with F : e e A MA → B MB , of the form M → A Hom A (C, M), where C is an (A , B )-bimodule e op e (A = A⊗A is the enveloping algebra of A, and B is the enveloping algebra of B). 2 (id ⊗ id ), Explicitly, given such an (F, F 2 , F 0 ), the coproduct for C is C = FC,C C B C 0 ∗ ∗ 0 the counit is εC = F (1B ), and the map ι : B → C is ι = F . 3.28. Dual ring of the Sweedler coring. Let B → A be an algebra map and let C be the associated Sweedler coring 3.10. Using the isomorphism HomA (A ⊗B A, A) End B (A), the right dual ring C ∗ is identiﬁed with the opposite of the endomorphism ring End B (A). Similarly, ∗ C is identiﬁed with the endomorphism ring B End(A) (both endomorphism rings with product given by the composition). 3.29. Dual ring of the comatrix coring. Let P be a (B, A)-bimodule such that P is ﬁnitely generated and projective as a right A-module, and let C = P ∗ ⊗B P be the associated comatrix coring 3.11. Using the isomorphisms A Hom(P

∗

⊗B P, A) B Hom(P, A Hom(P ∗ , A)) B End(P),

the left dual ring ∗ C is identiﬁed with the endomorphism ring B End(P). Similarly, C ∗ is identiﬁed with the opposite of the endomorphism ring End B (P ∗ ). 3.30. The ψ-twisted convolution algebra. Let (A, C)ψ be an entwining structure and let C = A ⊗ C be the associated A-coring; see 3.20, 3.21. With the isomorphism A Hom(A ⊗ C, A)

Homk (C, A),

the algebra ∗ C can be identiﬁed with the k-module of k-linear maps C → A with the product f ∗ψ g(c) = g(c(2) )ψ f(c(1) ψ ),

ψ

where ψ(c ⊗ a) = ψ aψ ⊗ cψ is the notation for the entwining map. The algebra Homk (C, A) with the above product (and unit 1A ◦ εC ) is known as a ψ-twisted convolution algebra. 3.31. Dual coring theorem [178, Theorem 3.7]. Let φ : A → S be an antialgebra map. View S as an A-bimodule with multiplications given by asb := φ(b)sφ(a), for all s ∈ S, a, b ∈ A. Then

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(1) If S is a ﬁnitely generated projective right A-module, then there is a unique A-coring structure on S ∗ = HomA (S, A) whereby the evaluation map β : S → ∗ (S ∗ ) = A Hom(HomA (S, A), A),

s → [σ → σ(s)],

is an algebra isomorphism. Here, ∗ (S ∗ ) has an algebra structure as in 3.25. Furthermore, β ◦ φ = ιL , where ιL is as in 3.25(2). (2) If S is a ﬁnitely generated projective left A-module, then there is a unique A-coring structure on ∗S = A Hom(S, A) whereby the evaluation map β : S → (∗S)∗ = HomA (A Hom(S, A), A),

s → [σ → σ(s)],

is an algebra isomorphism. Here, (∗S)∗ has an algebra structure as in 3.25. Furthermore, β ◦ φ = ιR , where ιR is as in 3.25(1). Sketch of proof. The coproduct in C = S ∗ is deﬁned as the unique map C : C → ) = c(s s). Explicitly, C⊗A C such that, for all s⊗A s ∈ S⊗A S and c ∈ C, C (c)(s⊗A s i i ∗ if {s ∈ S, c ∈ S } is a ﬁnite dual basis for S, then C (c) = i c(si −) ⊗A ci . The counit of C is deﬁned as εC : c → c(1S ). 4. Comodules Modules represent rings. Similarly, to represent corings one studies comodules. In this section, we describe the main properties and examples of comodules. 4.1. Deﬁnition Here, we give the deﬁnition and basic properties of comodules. 4.1. Comodules of a coring. Let C be an A-coring and let D be a B-coring. (1) A right C-comodule is a pair (M, M ), where M is a right A-module, and M : M → M ⊗A C is a right A-module map making the following diagrams M M

M ⊗A C

M

/ M ⊗A C M⊗A C

/ M ⊗ A C ⊗A C ,

M ⊗A C

M

/ M ⊗A C M H HH HH H M⊗A εC

HHH $ M ⊗A A

commute. The map M is called a right coaction. A morphism of right C-comodules (or a right C-colinear map) (M, M ) → (N, N ) is a right A-module map f : M → N such that (f ⊗A C) ◦ M = N ◦ f. The category with objects right C-comodules and morphisms right C-colinear maps is denoted by MC ; k-modules of morphisms in this category are denoted by HomC (M, N).

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(2) A left C-comodule is a pair (M, M), where M is a left A-module, and M : M → C ⊗A M is a left A-module map making the following diagrams M M

C ⊗A M

M

C⊗A

/ C ⊗A M

M

C ⊗A M

/ C ⊗A C ⊗ A M ,

M

/ C ⊗A M M H HH HH H εC ⊗A M

HHH $ A ⊗A M

commute. The map M is called a left coaction.Amorphism of left C-comodules (or a left C-colinear map) (M, M) → (N, N) is a left A-module map f : M → N such that (C ⊗A f ) ◦ M = N ◦ f. The category with objects left C-comodules and morphisms left C-colinear maps is denoted by C M; k-modules of morphisms in this category are denoted by C Hom(M, N). (3) A (C, D)-bicomodule is a triple (M, M , M), where M is an (A, B)-bimodule, (M, M ) is a right D-comodule, and (M, M) is a left C-comodule such that the right coaction M is left C-colinear or, equivalently, the left coaction M is right D-colinear. Morphisms or (C, D)-bicolinear maps are (A, B)-bimodule maps, which are at the same time left C-colinear and right D-colinear. The category of (C, D)-bicomodules is denoted by C MD ; morphisms are denoted by C HomD (M, N ). In case D = C one simply talks about C-bicomodules. Whatever is said about right comodules can be equivalently said about left comodules (this is known as a left-right symmetry). Thus, in this chapter, we restrict our attention mainly to right comodules. 4.2. The Sweedler–Heyneman notation. For a right C-comodule (M, M ), the coaction on elements m ∈ M is denoted by the Sweedler–Heyneman notation: m(0) ⊗A m(1) ∈ M ⊗A C. M (m) = Similar to coproducts, the coassociative law implies that m(0) ⊗A m(1)(1) ⊗A m(1)(2) = m(0)(0) ⊗A m(1)(1) ⊗A m(1) , and therefore one can relabel the Sweedler–Heyneman indices and write (M ⊗A C ) ◦ M (m) = (M ⊗A C) ◦ M (m) = m(0) ⊗A m(1) ⊗A m(2) . 4.3. The image of the coaction. If (M, M ) is a right C-comodule, then (M ⊗A C, M ⊗A C ) is also a right C-comodule. Furthermore, M is a C-comodule map. Since M has a retraction M⊗A εC (i.e. (M⊗A εC )◦M = M), M is a monomorphism of right C-comodules. Thus (M, M ) (M (M), M ⊗A C |M (M) ) as C-comodules.

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4.4. Factorizable morphisms of comodules. Let (M, M ), (N, N ), (P, P ) be right C-comodules and let f : M → N, g : N → P be A-module maps such that g ◦ f is a comodule morphism. If f is colinear, then P ◦ g ◦ f = (g ◦ f ⊗A C) ◦ M = (g ⊗A C) ◦ N ◦ f. Thus if f is an epimorphism of comodules, then g is a right C-comodule map. In particular, if a C-colinear map f has a k-linear (hence A-linear) inverse f −1 , then f −1 is necessarily a C-colinear map (i.e. f is an isomorphism of comodules). 4.5. The regular bicomodule. (C, C , C ) is a C-bicomodule, that is, C can be viewed as a right C-comodule and as a left C-comodule with coaction given by the coproduct C . This is known as a regular C-comodule. 4.6. Comodules ﬁnitely generated and projective as modules. Let (P, P ) be a right C-comodule. Assume that P is ﬁnitely generated and projective as a right A-module, and let {ei ∈ P, πi ∈ P ∗ = HomA (P, A)} be its ﬁnite dual basis. View P ∗ as a left A-module with the multiplication (af )(x) = af(x). Then P ∗ is a left C-comodule with the coaction P∗ (f ) = f(ei (0) )ei (1) ⊗A πi . i

4.7. Comodules as coalgebras. In view of 3.5, C is a coring if and only if − ⊗A C : MA → MA is a comonad. The category of − ⊗A C-coalgebras is isomorphic to the category of right C-comodules. Similarly, the category of left C-comodules is isomorphic to the category of coalgebras of the comonad C ⊗A − : A M → A M. 4.8. The deﬁning adjunctions [106, Proposition 3.1]. Since an A-coring C is given by comonads, whose coalgebras coincide with comodules, with every A-coring C, one can associate two pairs of adjoint functors between categories of A-modules and C-comodules. (1) The forgetful functor (−)A : MC → MA is a left adjoint of the tensor functor − ⊗A C : MA → MC ,

N → (N ⊗A C, N ⊗A C ).

The unit and the counit of this adjunction are η(M,M ) = M : M → M ⊗A C,

ψN = N ⊗A εC : N ⊗A C → N,

for all right C-comodules (M, M ) and right A-modules N. Consequently, there is an isomorphism (natural in N ∈ MA , (M, M ) ∈ MC ), M,N : HomA (M, N) → HomC (M, N ⊗A C),

M,N (g) = (g⊗A C)◦M .

The inverse of M,N is −1 M,N (f ) = (N ⊗A εC ) ◦ f . (2) The forgetful functor A (−) : C M → A M is a left adjoint of the tensor functor C ⊗A − : A M → C M,

N → (C ⊗A N, C ⊗A N).

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The unit and the counit of this adjunction are η(M,M) = M : M → C ⊗A M,

ψN = εC ⊗A N : C ⊗A N → N,

for all left C-comodules (M, M) and left A-modules N. Consequently, there is an isomorphism (natural in N ∈ A M, (M, M) ∈ C M), M,N : A Hom(M, N) → C Hom(M, C ⊗A N ),

M,N (g) = (C⊗A g)◦M.

The inverse of M,N is −1 M,N (f ) = (εC ⊗A N ) ◦ f . 4.9. Coinvariants. Let (M, M ) be a right C-comodule. Given any right C-comodule (N, N ), the k-module HomC (M, N ) is called the coinvariants of N with respect to M or M-coinvariants of N. The M-coinvariants or, simply, coinvariants of M are thus equal to the endomorphism ring S = HomC (M, M ) = End C (M ). The product in S is given by the composition of maps, that is, ss (m) = s(s (m)), for all s, s ∈ S, m ∈ M. The endomorphism ring S acts on M from the left by the evaluation, that is, for all s ∈ S and m ∈ M, sm := s(m). By the construction of S, the coaction M is left S-linear. Obviously, S is a subalgebra of the right A-module endomorphism ring End A (M ). 4.10. The coinvariants of the regular comodule. Let (M, M ) be a right C-comodule. In view of the hom-tensor relations 4.8, the M-coinvariants of the regular C-comodule (C, C ) are isomorphic to the right dual module M ∗ , that is, HomC (M, C) HomC (M, A ⊗A C) HomA (M, A) = M ∗ . In particular, the endomorphisms of the right regular C-comodule (C, C ) are isomorphic to the right dual module C ∗ . When the algebra structure of End C (C) is ported through this isomorphism to C ∗ , it coincides with the opposite algebra structure of the dual ring C ∗ as described in 3.25. 4.11. The coinvariants adjunction. For a right C-comodule (M, M ), let S = End C (M) be the ring of M-coinvariants of M; see 4.9. For any right C-comodule (N, N ), the M-coinvariants of N are a right S-module with the multiplication fs := f ◦ s. Thus the M-coinvariants deﬁne a functor HomC (M, −) : MC → MS . Since this is the hom-functor, it has the left adjoint, the tensor functor − ⊗S M : MS → MC ,

X → (X ⊗S M, X ⊗S M ).

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Explicitly, the counit and unit of adjunction are ψN : HomC (M, N ) ⊗S M → N, ηX : X →

HomC (M, X ⊗S

M),

f ⊗S m → f(m), x → [m → x ⊗S m];

see 2.9. In other words, for all (N, N ) ∈ MC and X ∈ MS , there is an isomorphism [natural in (N, N) and X] HomC (X ⊗S M, N) → HomS (X, HomC (M, N)), δ → [x → δ(x ⊗S −)] , with the inverse ϕ → [x ⊗S m → ϕ(x)(m)]. This adjoint pair of functors has been one of the main objects of studies in the coring theory in recent years. We refer to it as the coinvariants adjunction. 4.12. Group-like elements. The (regular) right (or left) A-module A is a right (or left) comodule of an A-coring if and only if there exists an element g ∈ C such that C (g) = g ⊗A g,

εC (g) = 1A .

An element g satisfying the above properties is called a group-like element. Given a group-like element g, a right C-coaction on A is deﬁned by A : A → A ⊗A C C,

a → ga,

while a left C-coaction is A : A → C ⊗A A C, a → ag. For example, the Sweedler coring C = A ⊗B A associated to an algebra map B → A (cf. 3.10) has a group-like element 1A ⊗B 1A . Thus, A is a right C-comodule with the coaction a → 1A ⊗B a and a left C-comodule with the coaction a → a ⊗B 1A . 4.13. Group-like elements and differential graded algebras. Any group-like element g in an A-coring C gives rise to a semifree differential graded algebra. This construction provides one with the inverse to the construction of corings in 3.16. Since a coring associated to a semifree differential graded algebra has a group-like element (the “indeterminate” g in 3.16), this establishes a bijective correspondence between corings with a group-like element and semifree differential graded algebras [162]. Given an A-coring C with a group-like element g, one deﬁnes a semifree differential graded algebra as follows: 1 = ker εC ,

n+1 = 1 ⊗A n ,

and the multiplication is given by the tensor product [i.e. is the tensor algebra = TA (ker εC )]. The differential is deﬁned by d(a) = ga − ag,

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for all a ∈ A, and for all c1 ⊗A · · · ⊗A cn ∈ (ker εC )⊗A n , d(c1 ⊗A · · · ⊗A cn ) = g ⊗A c1 ⊗A · · · ⊗A cn + (−1)n+1 c1 ⊗A · · · ⊗A cn ⊗A g n (−1)i c1 ⊗A · · · ⊗A ci−1 ⊗A C (ci ) ⊗A ci+1 ⊗A · · · ⊗A cn . + i=1

The differential algebra associated to the Sweedler coring A ⊗B A and a group-like element 1A ⊗B 1A coincides with the algebra of B-relative universal differential forms on A (cf. [80]). 4.14. The g-coinvariants of A. Suppose that A is a right (or left) C-comodule with the coaction given by a group-like element g. Let (M, M ) be a right C-comodule. Then the isomorphism of k-modules M HomA (A, M ),

m → [a → ma],

allows one to identify A-coinvariants of M with the k-module Mg = {m ∈ M | M (m) = m ⊗A g}. Mg is called the g-coinvariants of M. In particular, the g-coinvariants of A are the subalgebra Ag of A deﬁned by Ag = {b ∈ A | gb = bg}. As Ag is isomorphic to End C (A) and Mg is isomorphic to HomC (A, M ), the g-coinvariants Mg form a right Ag -module. M is a left Ag -module and M is a left Ag -linear map. 4.15. The cotensor product. Let (M, M ) be a right comodule and let (N, N) be a left comodule over an A-coring C. The cotensor product M2C N is the k-module deﬁned by the equalizer M2C N

/ M ⊗A N

M ⊗A N M⊗A

N

/

/ M ⊗ A C ⊗A N .

Put differently, M2C N is the kernel of the k-linear map ωM,N = M ⊗A N −M ⊗A N. Given a right A-module M , left A-module N , and module maps f : M → M , g : N → N , we write f C g for the restriction of f ⊗A g to MC N. In case M , N are comodules and f , g are comodule maps, f C g factors through the canonical inclusion M C N ⊆ M ⊗A N and hence is considered as a map MC N → M C N . If either (M, M ) or (N, N) is the regular comodule (C, C ), then M2C C = M (M) M,

C2C N = N(N) N,

with the isomorphisms given by MC εC and εC C N, respectively. 4.16. Coinvariants and the cotensor product. Take any right C-comodules (M, M ) and (N, N ) and assume that M is ﬁnitely generated and projective as a right ∗ A-module. Let M ∗ = HomA (M, A) be the dual module, and let (M ∗ , M ) be the

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induced left C-comodule as in 4.6. Then the k-module isomorphism N ⊗A M ∗ HomA (M, N ),

n ⊗A f → [m → nf(m)],

restricts to the isomorphism N2C M ∗ HomC (M, N ). Thus, in particular, in case (N, N ) = (M, M ), the coinvariants S = End C (M ) of M can be identiﬁed with the cotensor product M2C M ∗ . 4.17. The tensor–cotensor relations. In general, if (M, M ) is a right C-comodule and (N, N , N) is a (C, D)-bicomodule, the coaction M ⊗A N on M ⊗A N does not restrict to a coaction in MC N. It does restrict if the map ωM,N = M ⊗A N −M⊗A N is D ⊗B D-pure as a right B-module map. The cotensor product is not an associative operation and it does not commute with the tensor product. Given algebras A and B and an A-coring C, if (M, M ) is a right C-comodule, (N, N) is a (C, B)-bicomodule, and V is a left B-module, then there is a k-linear map τM,N,V : (M2C N) ⊗B V → M2C (N ⊗B V ), which is not an isomorphism unless ωM,N = M ⊗A N − M ⊗A N is V -pure. Here, N ⊗B V is a left C-comodule by N ⊗B V . Thus (M2C N) ⊗B V M2C (N ⊗B V ) if, for example, V is a ﬂat left B-module or (M, M ) is a (C, A)-injective comodule [see 5.1 for the deﬁnition of (C, A)-injectivity]. For a detailed discussion of the cotensor product see, [101] and [53] (including erratum). 4.18. Coinvariants of a coring morphism [133]. Let φ : D → C be a morphism of A-corings. Then (D, (D ⊗A φ) ◦ D , (φ ⊗A D) ◦ D ) is a C-bicomodule. Set B = C HomC (D, C). B is an algebra with unit φ and product, for all b, b ∈ B, d ∈ D, bb (d) = εC (b(d (1) ))b (d (2) ) = b(d (1) )εC (b (d (2) )). The algebra B is called an algebra of coinvariants of the A-coring morphism φ. Given an A-coring C and a right C-comodule (P, P ) that is ﬁnitely generated projective as a right A-module, consider the comatrix coring D = P ∗ ⊗ P; see 3.11. Then f ⊗ x → f(x(0) )x(1) , φ : D = P ∗ ⊗ P → C, is an A-coring morphism. The induced (by φ) C-bicomodule structure on D comes ∗ ∗ out as (D, P ∗ ⊗ P , P ⊗ P), where P is a left C-coaction as in 4.6. Since P is a ﬁnitely generated and projective A-module, A HomA (P ∗ ⊗ P, A) End A (P). This restricts to an isomorphism of algebras C

HomC (P ∗ ⊗ P, C) End C (P),

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261

(note that C HomC (P ∗ ⊗ P, C) ⊆ A HomA (P ∗ ⊗ P, A) by the deﬁning adjunction). Thus, the coinvariants of the A-coring morphism φ : P ∗ ⊗ P → C coincide with the P-coinvariants of P. 4.2. Examples Here, we describe comodules of examples of corings discussed in Section 3.2. In this way, we discover objects (and categories) studied in the precoring literature by different means. 4.19. Comodules of the trivial A-coring. Since both coproduct and counit of the trivial A-coring A are identity maps to be a right comodule of a trivial A-coring, A is the same as to be a right A-module. Thus MA = MA and A M = A M. In particular, as observed in 4.9, given an A-coring C and a right C-comodule (M, M ), the coaction M is left linear over the coinvariants S = End C (M). Viewing S as a trivial S-coring, we can thus say that M is an (S, C)-bicomodule. 4.20. Descent data. Let C = A ⊗B A be the Sweedler coring associated to an algebra map B → A; see 3.10. Right C-comodules are in bijective correspondence with descent data (cf. [104]). The latter are deﬁned as pairs (M, f ), where M is a right A-module and f : M → M ⊗B A is a right A-module morphism such that writing f(m) = i mi ⊗B ai , for any m ∈ M, f(mi ) ⊗B ai = mi ⊗B 1A ⊗B ai , mi ai = m. i

i

i

This bijective correspondence is obtained simply by identifying M ⊗A A ⊗B A with M ⊗B A. A morphism of descent data (M, f ) → (M , f ) is a right A-module map φ : M → M such that f ◦ φ = (φ ⊗B A) ◦ f . The category of descent data is denoted by Desc(A|B). This category has been introduced in [125] (for commutative rings) and [74] (for general rings). The identiﬁcation of right C-comodules with descent data is compatible with the deﬁnitions of morphisms in respective categories. Thus there is an isomorphism of categories MA⊗B A ≡ Desc(A|B); see [37], [68, Section 4.8]. 4.21. Comodules in a comatrix coring context. Consider a comatrix coring context (A, B, N, M, σ, τ) and the associated coring C = N ⊗B M; see 3.13, 3.14. Recall that B is a ﬁrm (nonunital) algebra, and write ∇M : M → B ⊗B M for the inverse of the multiplication map B ⊗B M → M and ∇N : N → N ⊗B B for the inverse of the multiplication map N ⊗B B → B. Then M is a right C-comodule with the coaction M : M → M ⊗A N ⊗B M,

M = (τ ⊗B M ) ◦ ∇M ,

and N is a left C-comodule with the coaction N

: N → N ⊗B M ⊗A N,

N

= (N ⊗B τ) ◦ ∇N .

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4.22. Comodules and (ﬁnite) comatrix corings. A comatrix A-coring associated to a (B, A)-bimodule P, ﬁnitely generated and projective as a right A-module (cf. 3.11), is a special case of a coring associated to a comatrix coring context. Thus both P and P ∗ are C-comodules, with the right and left coactions, respectively, P : P → P ⊗A P ∗ ⊗B P, P∗

: P ∗ → P ∗ ⊗B P ⊗A P ∗ ,

x → e ⊗B x, π → π ⊗A e,

where e is the dual basis element.

n 4.23. Flat connections [38]. Let = ∞ n=0 be a semifree differential algebra • •+1 with a differential d : → . Set A = 0 , and let C = Ag ⊕ 1 be the associated A-coring; see 3.16. Take a right A-module M. Any k-linear map M : M → M ⊗A C = M ⊗A Ag ⊕ M ⊗A 1 such that (M ⊗A εC ) ◦ M = M can be identiﬁed with a map ∇ : M → M ⊗A 1 by M (m) = m ⊗A g + ∇(m). The map M is a right A-module map if and only if, for all m ∈ M, a ∈ A, ∇(ma) = ∇(m)a + md(a). A map ∇ : M → M ⊗A 1 satisfying this condition is known as the connection in M with respect to (cf. [21, 2.8 Deﬁnition], [76]). The map M is coassociative if and only if (∇ ⊗A 1 ) ◦ ∇ + (M ⊗A d ) ◦ ∇ = 0. The map F := (∇ ⊗A 1 ) ◦ ∇ + (M ⊗A d ) ◦ ∇ is known as a curvature of ∇, and a connection with vanishing curvature is called a ﬂat connection. Thus, right C-comodules are in bijective correspondence with modules with ﬂat connections with respect to . 4.24. Entwined modules [36] [37, Proposition 2.2]. Let (A, C)ψ be an entwining structure and let C = A ⊗ C be the associated A-coring; see 3.20, 3.21. Using the identiﬁcation M ⊗A C = M ⊗A A ⊗ C M ⊗ C, one identiﬁes right C-comodules with triples (M, M , M ), where (M, M ) is a right A-module (with A-multiplication M : M ⊗ A → M) and (M, M ) is a right C-comodule such that M⊗A

M ⊗A

M

M

M

/ M⊗C⊗A PPP PPPM⊗ψ PPP PPP ( M⊗A⊗C nnn nnn n n n wnnn M ⊗C / M⊗C

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is a commutative diagram. Any such triple (M, M , M ) is called an entwined module. A morphism of entwined modules is a right A-linear map, which is also right C-colinear. Entwined modules and their morphisms form a category denoted by C (ψ). Again, through the identiﬁcation M ⊗ C M ⊗ C, morphisms of entwined MA A C (ψ). modules can be identiﬁed with right C-colinear maps. Hence MC ≡ MA 4.25. Weak entwined modules [56], [37, Proposition 2.3]. Let (A, C, ψ) be a weak entwining structure, and let C be the associated A-coring; see 3.22, 3.23. By similar arguments to those in 4.24, one identiﬁes right C-comodules with right weak entwined modules. The latter are deﬁned as right A-modules and right C-comodules satisfying the same compatibility condition as in the diagram in 4.24. 4.3. The category of comodules In this section, we discuss properties of the category of comodules, and in particular, we describe how properties of C as an A-module affect the structure of the category MC . Although many notions appearing in this section are explained in Section 2, some additional background in category theory might be required to understand some of the topics discussed here. 4.26. Comodules form a k-linear category. Since every hom-set HomC (M, N) is a k-module and the composition of morphisms in MC is k-linear, the category MC is a k-linear (hence also preadditive) category. 4.27. Cokernels. Consider any morphism f : (M, M ) → (N, N ) in MC . Since f is a morphism of right A-modules, it has a cokernel g : N → L in the category of right A-modules. Furthermore, f is a morphism of C-comodules; hence, g ◦ f = 0 implies (g ⊗A C) ◦ N ◦ f = 0. By the universality of cokernels, there is a unique A-module map L : L → L ⊗A C such that L ◦ g = (g ⊗A C) ◦ N . The coassociativity and counitality of N (together with the fact that g is surjective) imply the coassociativity and counitality of L . Hence (L, L ) is a right C-comodule. By construction, g is a comodule map. By the universality of cokernels in MA , for any h : (N, N ) → (P, P ) such that h ◦ f = 0, there is a unique A-module map h : L → P such that h ◦ g = h. Since both h and g are comodule maps and g is an epimorphism, also h is a comodule map by 4.4. This proves the universality of (L, L ). Hence (L, L ) is a cokernel of the right C-comodule morphism f . Thus the category MC has cokernels. 4.28. Kernels. Consider any morphism f : (M, M ) → (N, N ) in MC . Since f is a morphism of right A-modules, it has a kernel h : K → M in the category of right A-modules. By tensoring the deﬁning sequence of the kernel with C, one obtains the following commutative diagram 0

/ K

h

/ M M

0

/ K ⊗A C h⊗A C / M ⊗A C

f

/ N N

f ⊗A C / N ⊗A C .

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The top sequence is exact by the deﬁnition of a kernel of an A-module morphism. However, since the tensor functor does not necessarily preserve monomorphisms, the bottom sequence need not be exact. The exactness of the bottom sequence is equivalent to saying that f is C-pure as a right A-module morphism. Thus, if the bottom sequence is exact, the diagram can be extended commutatively to give a map K : K → K ⊗A C; see 4.27. To check that the map K is coassociative, one needs the exactness of the following sequence: 0

/ K ⊗ A C ⊗A C

h⊗A C⊗A C

/ M ⊗ A C ⊗A C

f ⊗A C⊗A C

/ N ⊗A C ⊗A C.

This means that one needs to assume that f is C ⊗A C-pure as a right A-module map. By the counitality of the coproduct, this implies that f is C-pure. Thus, if f is C ⊗A C-pure, then (K, K ) is a right C-comodule. The pair (K, K ) is the kernel of the comodule morphism f . If C is a ﬂat left A-module, then the tensor functor − ⊗A C : MA → Mk is exact. This implies that every right A-module map is C ⊗A C-pure. Consequently, every C-comodule morphism has a kernel. Thus, the category of right C-comodules has kernels provided C is ﬂat as a left A-module. Mλ 4.29. Coproducts in the category of comodules. Let {(M λ , )}λ∈ be a family of right C-comodules. Consider the right A-module M = λ∈ Mλ in MA and deﬁne the right A-module map M : Mλ = M → M ⊗A C = (Mλ ⊗A C), M |Mλ = Mλ . λ∈

λ∈

Mλ

Since each of the is coassociative, so is their coproduct M : M → M ⊗A C. M Hence, (M, ) is a right C-comodule such that the inclusions Mλ → M are right C-comodule morphisms. This is the coproduct of the family of right C-comodules {(Mλ , Mλ )}λ∈ . 4.30. Comodules of a ﬂat coring [96, Proposition 1.2]. As a consequence of the deﬁning adjunctions 4.8, the forgetful functor (−)A : MC → MA has a right adjoint; hence it preserves epimorphisms, that is, every epimorphism of right C-comodules is a right A-linear epimorphism. On the other hand, (−)A : MC → MA preserves monomorphisms if and only if C is ﬂat as a left A-module. This leads to the equivalence of the following statements: (a) C is ﬂat as a left A-module; (b) MC is an Abelian category and the forgetful functor (−)A : MC → MA is exact (i.e. it preserves exact sequences); (c) MC is a Grothendieck category and (−)A : MC → MA is an exact functor. 4.31. A Grothendieck category of comodules over a nonﬂat coring [96, Example 1.1]. The following two examples show that ﬂatness of C as a left A-module is not necessary for the category of comodules to be Abelian (Grothendieck).

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(1) Take k-algebras R, S, and an (R, S )-bimodule M, and let

R M R M A= , C= . 0 S 0 0 A is a (matrix) algebra, and C is an ideal in A. C has a coproduct given by the isomorphism C C ⊗A C and a counit given by the inclusion C ⊆ A. Right C-comodules can be identiﬁed with triples (X, Y, θ), where X is a right R-module, Y is a right S-module, and θ : X ⊗R M → Y is an isomorphism of right S-modules. The functor (X, Y, θ) → X establishes an equivalence of categories MC and MR . The latter is a Grothendieck category hence so is the former. Yet C is not necessarily a ﬂat left A-module. (2) Even more explicit example of a nonﬂat coring, whose category of comodules is Abelian (Grothendieck), is constructed in [81, Remark 2.2.9] (cf. [34, Exercise 4.8.13]), by using the correspondence between comodules of the Sweedler coring and the category of descent data; see 4.20. Consider a monomorphism of commutative rings f : B → A. By the Joyal–Tierney theorem (see [137, Theorem] for the commutative case and [55, Proposition 2.3], [59, Theorem 2.7] for noncommutative versions), the functor − ⊗B A : MB → Desc(A|B) ≡ MA⊗B A is an equivalence if and only if f is a pure morphism of B-modules (i.e. N ⊗B f is a monomorphism for all B-modules N). Take B = Z, A = Z × Z/2Z, and f : n → (n, [n]). Then A is not a ﬂat B-module (the map i ⊗B A, where i : 2Z → Z is the inclusion, has a nontrivial kernel), and hence A ⊗B A is not a ﬂat left A-module. Yet, f is pure as a Z-module map, as it has a Z-module retraction (n, [m]) → n. Therefore, MA⊗B A is equivalent to a Grothendieck category MZ , hence, MA⊗B A is a Grothendieck category. 4.32. Monomorphisms and products. As a consequence of the deﬁning adjunctions 4.8, the functor − ⊗A C : MA → MC has a left adjoint; hence, it preserves monomorphisms, that is, if f is a monomorphism of right A-modules, then f ⊗A C is a monomorphism of right C-comodules. For the same reason, given a family of right A-modules {Mλ }λ∈ , the comodule (( Mλ ) ⊗A C, ( Mλ ) ⊗A C ) is the product of the comodules (Mλ ⊗A C, Mλ ⊗A C ). 4.33. Comodules and modules of the dual ring. Let C be an A-coring and let B = ∗ C be the dual ring deﬁned in 3.25(2). Every right C-comodule (M, M ) is a left B-module with the multiplication, for all b ∈ B, m ∈ M, bm = m(0) b(m(1) ). Furthermore, every right C-comodule map is a left B-module map. This deﬁnes a functor MC → B M. Clearly, if two right C-comodule maps differ when viewed as left B-module maps, they differ as right C-comodule maps. Hence the functor MC → B M is faithful.

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4.34. Comodules of a locally projective coring. Following [99, 204], a left A-module P is said to be locally projective if every ﬁnitely generated submodule of P is a ﬁnitely generated projective module. Equivalently, P is a locally projective left A-module if, for any diagram of left A-modules with exact rows, 0

/ F

i

/ P g

M

f

/ N

/ 0,

where F is ﬁnitely generated, there exists h : P → M such that g ◦ i = f ◦ h ◦ i. Equivalently, P is a locally projective left A-module if and only if, for any left A-module N, the map αN : N ⊗A P → HomA (A Hom(P, A), N),

n ⊗A x → [f → nf(x)],

is injective; see [204, Theorem 2.1]. If P is a locally projective module, then P is a ﬂat module. Clearly, if P is projective as a left A-module, then P is locally projective. Thus, local projectivity is a property of a module located in-between ﬂatness and projectivity. For a detailed discussion of local projectivity, the reader is referred to [189]. In relation to corings, one refers to an A-coring C that is a locally projective left A-module as an A-coring satisfying the left α-condition; see [2] (these notions arise from studies of rationality of coalgebras over commutative rings [7, 100]). An A-coring C satisﬁes the left α-condition if and only if the category of right C-comodules is a full subcategory of the category of left modules over the dual ring B = ∗ C; see [197] (cf. [53, 19.3]). In other words, C is a locally projective module if and only if the functor MC → B M is full, that is, HomC (M, N) = B Hom(M, N), for any comodules (M, M ), (N, N ) ∈ MC . 4.35. Rational modules and the rational functor. Let C be an A-coring, which is locally projective as a left A-module, and let B = ∗ C. Forany left B-module M, r ∈ M is called a rational element if there exists an element i mi ⊗A ci ∈ M ⊗A C such that for all b ∈ B, br = i mi b(ci ). Since αM : M ⊗A C → HomA (B, M) in 4.34 is assumed to be injective, such an element i mi ⊗A ci is uniquely determined. The set of all rational elements of M is a (left B-, hence right A-) submodule of M, C which is called the rational submodule of M, and is denoted by Rat (M). Since, for any r the corresponding i mi ⊗A ci is uniquely determined, there is a right A-linear map Rat C (M ) → M ⊗A C, r → mi ⊗A ci . i C Rat (M)

One can check that all the mi are in and that this map is coassociative and C counital, thus giving rise to a coaction Rat (M) on Rat C (M ). This deﬁnes a functor: Rat C : B M → MC , known as the rational functor.

M f :M→N

C

→ (Rat C (M), Rat (M) ),

→ f |RatC (M) : Rat C (M) → Rat C (N),

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Rat C (M) = M if and only if the left B action on M is obtained from aCC-coaction that is, (M, M ) ∈ MC , and bm = m(0) b(m(1) ). The equality Rat (M) = M holds for all left B-modules M if and only if C is ﬁnitely generated as a left A-module (and thus is ﬁnitely generated and projective). For more details on rational modules including their description in terms of pairings and their topological aspects, the reader is referred to [2, 62, 69].

M ,

4.36. Comodules as the Wisbauer σ-category [53, 19.3]. An A-coring C is locally projective as a left A-module if and only if MC is the smallest full Grothendieck subcategory of B M, which contains C. Such a category is denoted by σ[C] and is called the Wisbauer σ-category. Equivalently, σ[C] is characterized as the full subcategory of B M consisting of all submodules of modules M, for which there is an epimorphism C () → M (i.e. σ[C] consists of C-subgenerated modules). The Wisbauer σ-category σ[C] contains large amount of module-theoretic information about C as a left B-module. For detailed studies of σ-categories, we refer to [195]. 4.37. Comodules of a ﬁnitely generated and projective coring. Let C be an A-coring and B = ∗ C the left-dual ring. The functor MC → B M described in 4.33 can be combined with the deﬁning functor − ⊗A C : MA → MC to give the functor − ⊗A C : MA → B M. On the other hand, since B is an Aop -ring, there is a restriction of scalar functors (−)B/A : MB → MA . Lemma. The functor − ⊗A C : MA → B M is a right adjoint of the restriction of scalar functors (−)B/A : B M → MA if and only if C is a ﬁnitely generated and projective left A-module. Sketch of proof. If − ⊗A C is a right adjoint of (−)B/A , then the dual basis element for C is ηB (1B ) ∈ B ⊗A C, where η is the unit of the adjunction. Conversely, if {ci ∈ C, bi ∈ B} is a dual basis for C, then the unit of the adjunction is deﬁned as ηM : M → M ⊗A C, m → i bi m ⊗A ci , and the counit is ψN = N ⊗A εC , for all M ∈ B M, N ∈ MA . If C is ﬁnitely generated and projective as a left A-module, then one can identify the category MC with the category B M. More precisely, the functor MC → B M is an isomorphism of categories with the inverse given on objects as follows. Let {ci ∈ C, bi ∈ B} be a ﬁnite dual basis for C. Any left B-module M is mapped to a right C-comodule (M, M ), with coaction m → bi m ⊗ A c i , M : M → M ⊗A C, i

where M is a right A-module by the left multiplication with ιL , that is, ma = ιL (a)m = εC (−a)m. Conversely, if MC ≡ B M, then, since there is the deﬁning adjoint pair of functors ((−)A = (−)B/A , −⊗A C), Lemma implies that C is a ﬁnitely generated and projective left A-module.

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4.38. Module properties of a coring and the structure of comodules. Summary. The impact of module properties of a coring on the category of its comodules can be summarized as the following three statements about an A-coring C and its dual ring B = ∗ C: (1) If C is ﬂat as a left A-module, then MC is a Grothendieck category. (2) MC is the smallest full Grothendieck subcategory of B M containing C if and only if C is a locally projective left A-module (hence a ﬂat left A-module). (3) MC coincides with the category B M if and only if C is a ﬁnitely generated and projective left A-module (hence a locally projective left A-module). For example, taking C = A⊗D A to be the Sweedler coring associated to an algebra map D → A, we obtain the following information about the category of descent data Desc(A|D) (cf. 4.20): (1) Desc(A|D) is a Grothendieck category provided A is ﬂat as a left D-module. (2) If A is locally projective as a left D-module, then Desc(A|D) is the smallest full Grothendieck subcategory of the category of left modules over the endomorphism ring B = D End(A) that contains A ⊗D A [where A ⊗D A is a left B-module via b(a ⊗D a ) = a ⊗D b(a )]. (3) If A is a ﬁnitely generated and projective as a left D-module, then Desc(A|D) ≡ D End(A) M. 5. Special types of corings and comodules In this section, we describe several classes of corings, which satisfy additional properties. Such corings arise in a natural way in discussions of derived functors of the cotensor product or from the analysis of ring extensions (coseparable corings, cosplit corings, and Frobenius corings) or from descent theory (Galois corings). The richness of the structure of a coring has a bearing on its category of comodules. We discuss this matter here. 5.1. Coseparable and cosplit corings 5.1. Injective comodules. A right C-comodule (M, M ) is called a (C, A)-injective comodule if, for every right C-comodule map i : N → L that is a section in MA , every diagram i /L (∗) NA AA AA A f AA M of right C-comodule maps can be completed commutatively by a right C-comodule map g : L → M. Equivalently, (M, M ) is a (C, A)-injective comodule if and only if there exists a right C-colinear map σ M : M ⊗A C → M such that σ M ◦ M = M. Indeed, since

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M : M → M ⊗A C is a section in MA with retraction M ⊗A εC : M ⊗A C → M, the diagram M

MB BB BB B M BB

/L

M

meets all the above requirements. Hence, if (M, M ) is (C, A)-injective, then there is a map σ M as required. Conversely, if there is such a map, then the diagram (∗) can be completed by g = σ M ◦ (f ⊗A C) ◦ (p ⊗A C) ◦ L , where p : L → N is a right A-linear retraction of i. Still equivalently (M, M ) is a (C, A)-injective comodule if and only if every right C-comodule map M → L, which is a section in MA , is a section in MC . To see this, simply set N = M and f = M in the diagram (∗). Since C has a right C-colinear retraction εC ⊗A C, the regular right C-comodule (C, C ) is (C, A)-injective. More generally and by the same token, for any right A-module X, (X ⊗A C, X ⊗A C ) is a (C, A)-injective comodule. 5.2. Right (C, A)-semisimple corings. An A-coring C whose all right comodules are (C, A)-injective is called a right (C, A)-semisimple coring. 5.3. Coseparable corings. Coseparable corings are an example of right (and left) (C, A)-semisimple corings. An A-coring C is called a coseparable coring [106] if the coaction C has a Cbicolinear retraction, that is, if there exists an A-bimodule map π : C ⊗A C → C such that π ◦ C = C, and the following diagrams are commutative C ⊗A C π

C

C⊗A C

C

/ C ⊗A C ⊗A C

π⊗A C

/ C ⊗A C ,

C ⊗A C π

C

C ⊗A C

C

/ C ⊗A C ⊗A C

C⊗A π

/ C ⊗A C .

The existence of such a map π is equivalent to the existence of an A-bimodule map δ : C ⊗A C → A such that for all c, c ∈ C, δ(c ⊗A c (1) )c (2) , δ(c(1) ⊗A c(2) ) = εC (c). c(1) δ(c(2) ⊗A c ) = The correspondence is given by π → εC ◦ π,

δ → (δ ⊗A C) ◦ (C ⊗A C ) = (C ⊗A δ) ◦ (C ⊗A C).

A map δ satisfying the above properties is called a cointegral. Given a right comodule (M, M ) of a coseparable A-coring C, the right C-colinear map σ M = (M ⊗A δ) ◦ (M ⊗A C) is a retraction of M . This means that every right C-comodule of a coseparable coring C is (C, A)-injective, that is, a coseparable coring is (C, A)-semisimple.

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5.4. Coseparable corings as ﬁrm algebras [48]. Let C be a coseparable coring (with π and δ as in 5.3). Then C is a ﬁrm algebra with the product μC : C ⊗ C → C,

c ⊗ c → π(c ⊗A c ).

The inverse of the product map C⊗C C → C is given by the coproduct C (canonically projected to C ⊗C C). Furthermore, every right C-comodule (M, M ) is a ﬁrm right C-module with the multiplication, for all m ∈ M, c ∈ C, mc = m(0) δ(m(1) ⊗A c). The inverse ∇M : M → M ⊗C C of the multiplication map is given by M (projected canonically to M ⊗C C). 5.5. Cosplit corings. An A-coring C is called a cosplit coring if there exists an element e ∈ C A := {c ∈ C | ∀a ∈ A, ca = ac} such that εC (e) = 1A . Equivalently, one requires that the counit εC has an A-bimodule section. Such an element e is called a normalized integral in C. The name “cosplit” is motivated by the observation that a dual ring of a cosplit coring is a split extension. More precisely, an algebra map i : A → B is said to be a split extension [154] if it is an A-bimodule section. Let B = (∗ C)op be the opposite left dual ring of a cosplit A-coring C. Then, the algebra map ιL : A → B, a → εC (−a) is a split extension. Indeed, if e ∈ C A is a normalized integral, then the map E : B → A, f → f(e), is an A-bilinear retraction of ιL . 5.6. Separable bimodules [176] (cf. [65]). Let M be a (B, A)-bimodule. M is called a separable bimodule or B is said to be M-separable over A, provided the evaluation map M ⊗A B Hom(M, B) → B,

m ⊗A ϕ → ϕ(m),

is a retraction (i.e. a split epimorphism) of B-bimodules. The notion of a separable bimodule includes the notion of a separable extension and a split extension. an algebra map A → B isseparable if there exists Recall that ∈ (B ⊗ B)B such that an invariant e = b ⊗ b i A A i i bi bi = 1B ; see [109, i Deﬁnition 2]. This is equivalent to the requirement that the product map B ⊗A B → B is a retraction of B-bimodules. In case M = B, the dual module B Hom(B, B) can be identiﬁed with B, and then the evaluation map coincides with the product map. Hence, in the case of an algebra map A → B, B is a separable (B, A)-bimodule if and only if A → B is a separable extension. Furthermore, as shown in [176] (cf. [117, Theorem 3.1]), if M is a separable bimodule, then B → S := End A (M) is a split extension. Conversely, if a (B, A)bimodule M is such that M is ﬁnitely generated and projective as a right A-module, and B → S is a split extension, then M is a separable (B, A)-bimodule.

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5.7. Separable bimodules and comatrix corings [45]. Let P be a (B, A)-bimodule that is ﬁnitely generated and projective as a right A-module, and let C = P ∗ ⊗B P be the corresponding comatrix coring. (1) The (A, B)-bimodule P ∗ is a separable bimodule if and only if C is a cosplit A-coring. (2) If P is a separable bimodule, then C is a coseparable A-coring. (3) Let S = End A (P) and suppose that S is a faithfully ﬂat left or right B-module. If C is a coseparable A-coring, then P is a separable bimodule. 5.8. Separable functors. The notions of coseparable and cosplit corings are complementary to each other in the sense that the coseparablity and the cosplit property correspond to the same property of functors in the deﬁning adjunction. This common property is that of separability. The notion of a separable functor was introduced in [144]. Following the formulation in [159], a functor F : A → B is said to be separable if the transformation A(−, −) → B(F(−), F(−)), f → F(f ) is a split natural monomorphism. In case the functor F has a right adjoint G, the Rafael theorem [159, Theorem 1.2] asserts that F is separable if and only if the unit of adjunction is a natural section, while G is separable if and only if the counit of adjunction is a natural retraction. 5.9. Coseparable functors and corings [37, Theorem 3.3, Corollary 3.6]. Let C be an A-coring. (1) The forgetful functor (−)A : MC → MA is a separable functor if and only if C is a coseparable coring. (2) The functor − ⊗A C : MA → MC is a separable functor if and only if C is a cosplit coring. 5.2. Frobenius corings 5.10. Frobenius corings. An A-coring C is called a Frobenius coring if there exist an A-bimodule map η : A → C and a C-bicomodule map π : C ⊗A C → C yielding a commutative diagram C⊗A η

/ C ⊗A C C QQQQ QQQ QQCQ π η⊗A C QQQ QQQ Q (/ π C ⊗A C C. Equivalently, identifying π with the corresponding A-bimodule map δ = εC ◦ π (cf. 5.3) and η with an element e ∈ C A , C is a Frobenius coring if and only if there exist e ∈ C A and an A-bimodule map δ : C ⊗A C → A such that for all c, c ∈ C, c(1) δ(c(2) ⊗A c ) = δ(c⊗A c (1) )c (2) , δ(c⊗A e) = δ(e⊗A c) = εC (c).

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The pair (π, e) is called a Frobenius system, while the pair (δ, e) is called a reduced Frobenius system. Note that any Frobenius A-coring C is a ﬁnitely generated projective left and right A-module. In terms of the reduced Frobenius system (δ, e), and writing C (e) = e ⊗ e ¯ , a dual basis for the right A-module C is {ei , δ(¯ei ⊗A −)} and for the left i A i i A-module is {¯ei , δ(− ⊗A ei )}. 5.11. Frobenius functors and Frobenius corings. In [141, Theorem 2.1], it has been observed that an algebra extension is a Frobenius extension if and only if the extension of scalars functor has the same left and right adjoint. Following [67], a functor F is said to be a Frobenius functor if and only if it has the same left and right adjoint (such functors were termed strongly adjoint pairs in [141]). C is a Frobenius A-coring if and only if − ⊗A C : MA → MC is a Frobenius functor and if and only if (−)A : MC → MA is a Frobenius functor. 5.12. Frobenius corings and Frobenius extensions [39, Proposition 2.5, Theorem 2.7]. In 3.18, we have described a relationship between corings and Frobenius extensions. This description is completed by the following observations: (1) Let A → B be a Frobenius extension with a Frobenius element β and a Frobenius homomorphism E. Then B is a Frobenius A-coring with coproduct β (viewed as a B-bimodule map B → B ⊗A B), counit E, and a Frobenius system (π, 1B ), where π : B ⊗A B → B, b ⊗A b → bb . (2) Let C be a Frobenius A-coring with a Frobenius system (π, e). Then, (a) C is a k-algebra with product cc = π(c ⊗A c ) and unit 1C = e. (b) A → C, a → ae = ea is a Frobenius extension with a Frobenius element β = C (e) and Frobenius homomorphism E = εC . Thus, the notions of a Frobenius coring and a Frobenius extension are dual to each other descriptions of the same situation. 5.13. Frobenius bimodules. A (B, A)-bimodule M is said to be Frobenius if M is ﬁnitely generated and projective as a left B- and right A-module and B Hom(M, B) HomA (M, A) as (A, B)-bimodules; see [9, p. 261] or [116, Deﬁnition 2.1]. The endomorphism ring theorem (cf. [116, Theorem 2.5]) asserts that if M is a Frobenius (B, A)-bimodule, then B → End A (M) is a Frobenius extension. 5.14. Frobenius bimodules and comatrix corings [45, Theorem 3.7, Corollary 3.8]. Let P be a (B, A)-bimodule that is ﬁnitely generated and projective as a right A-module, and let C = P ∗ ⊗B P be the corresponding comatrix coring. (1) If P is a Frobenius bimodule, then C is a Frobenius A-coring. (2) Assume that P is ﬁnitely generated and projective as a left B-module. Let S = End A (P) and suppose that S is a faithfully ﬂat left or right B-module and that S Hom(P, S) ∼ = HomA (P, A) as (A, B)-bimodules. If C is a Frobenius A-coring, then P is a Frobenius bimodule.

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In particular, if there is a Frobenius algebra map B → A, then P = A is a Frobenius (B, A)-bimodule, and hence, the corresponding Sweedler coring A⊗B A is a Frobenius coring. 5.15. Towers of Frobenius extensions [39, Theorem 3.8]. Suppose that C is a Frobenius A-coring with a Frobenius system (π, e). Then by 5.12, e viewed as a map A → C is a Frobenius extension with the Frobenius element C (e) and the Frobenius homomorphism εC . Now 5.14 implies that the Sweedler C-coring C ⊗A C is Frobenius with the Frobenius system (C ⊗A εC ⊗A C, C (e)). Then C ⊗A C is a ring with unit C (e) and the product (c ⊗A c )(c ⊗A c ) = c ⊗A εC (π(c ⊗A c ))c , and the extension C : C → C ⊗A C is Frobenius by 5.12. The Frobenius element explicitly reads e(1) ⊗A e ⊗A e(2) , and the Frobenius homomorphism is π. Apply 5.12 to deduce that Sweedler’s C ⊗A C-coring (C ⊗A C) ⊗C (C ⊗A C) ∼ = C ⊗A C ⊗A C is Frobenius. Iterating this procedure we obtain the following Jones-like tower of corings and algebra maps A

e

/C

C

/ C ⊗A C

C⊗A e⊗A C

/ C ⊗A C ⊗A C

/ . . .,

in which for all k = 1, 2, . . ., C ⊗A k−1 → C ⊗A k is a Frobenius extension and C ⊗A k is a Frobenius C ⊗A k−1 -coring. 5.16. Categorical interpretation. Many of the properties of Frobenius corings can be derived by purely categorical methods. Given any monoidal category (M, , E), a Frobenius algebra in M is an object F such that (F, μF , 1F ) is an algebra in (M, , E) and (F, F , εF ) is a coalgebra in M such that (μF F ) ◦ (F F ) = F ◦ μF = (F μF ) ◦ (F F); see [142]. In particular, for any category A, the category End A := Fun(A, A) of functors A → A is a monoidal category (with tensor product given by the composition of functors). A Frobenius algebra in End A is simply a pair consisting of a monad (F, ν, η) and a comonad (F, δ, σ) such that νF ◦ F(δ) = δ ◦ ν = F(ν) ◦ δF . If R : A → B is a Frobenius functor with the same left and right adjoint L, F = LR is both a monad and comonad. Both structures are compatible making F a Frobenius algebra in End A. One easily ﬁnds that, for a Frobenius algebra F in End A, the categories of algebras of the monad (F, ν, η) and coalgebras of the comonad (F, δ, σ) are mutually isomorphic. On the other hand, a Frobenius algebra F in the monoidal category of A-bimodules (with tensor product ⊗A ) is the same as a Frobenius A-coring or a Frobenius extension A → F . By arguments similar to those in 3.5, this is equivalent to the statement that F ⊗A − is a Frobenius algebra in End A M. The results presented in this section

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are just an interplay between these different descriptions of a Frobenius algebra (in monoidal categories). For example, a (B, A)-bimodule P is Frobenius if and only if P ⊗A − : A M → B M is a Frobenius functor with (left = right) adjoint P ∗ ⊗B −; see [9, Exercise 21.8]. Thus, P ∗ ⊗B P ⊗A − is a Frobenius algebra in End A M; hence P ∗ ⊗B P is a Frobenius algebra in the category of A-bimodules, that is, it is a Frobenius A-coring. We refer the reader to [142] for more details. A very accessible expanded review of [142] can also be found in [12]. Finally, we would like to mention that Frobenius algebras (hence also Frobenius corings) play an important role in description of certain quantum ﬁeld theories. A review of recent developments in this area can be found in [168]. 5.17. Co-Frobenius corings [112]. An A-coring C is said to be a left co-Frobenius coring in case there is an injective morphism C → ∗ C of left A ⊗ ∗ C-modules [i.e. (∗ C, Aop )-bimodules, where ∗ C is a right Aop -module via ιL ; see 3.25]. C is a right co-Frobenius coring in case there is an injective morphism C → C ∗ of right A ⊗ C ∗ modules; see [130] for the case of coalgebras. C is called a left quasi-co-Frobenius coring provided that there are a set and an injective morphism C → (∗ C) of left A ⊗ ∗ C-modules. A Frobenius A-coring C is isomorphic to ∗ C as a left A ⊗ ∗ C-module and is isomorphic to C ∗ as a right A ⊗ C ∗ -module. Explicitly, if (δ, e) is a reduced Frobeisomorphism is θL : C → ∗ C, nius system for C, then the left A ⊗ ∗ C-module e(1) ξ(e(2) ). The right A ⊗ C ∗ -module c → [c → δ(c ⊗A c)], with inverse ξ → ∗ isomorphism is θR : C → C , c → [c → δ(c ⊗A c )], with inverse ξ → ξ(e(1) )e(2) . Thus, in particular, a Frobenius coring is left and right co-Frobenius. Recall that C is a Frobenius coring if and only if the functor − ⊗A C is Frobenius. Similar characterization exists also for (quasi)-co-Frobenius corings. A detailed discussion of (quasi)-co-Frobenius corings and related functors can be found in [112]. 5.18. Generalizations of Frobenius corings. Quasi-Frobenius corings are deﬁned in [72] in terms of quasi-Frobenius functors [141]. Let A, B be categories with ﬁnite direct sums (ﬁnite coproducts, e.g. Grothendieck categories). A functor L : B → A is said to be similar to R : B → A if there exist positive integers m, n and natural transformations α : L → Rn , α : Rn → L, β : R → Lm , β : Lm → R such that

α ◦ α = L,

β ◦ β = R.

Here, Rn (resp., Ln ) is a functor deﬁned on objects B as the direct sum of n-copies of R(B) [resp., L(B)]. A functor F : A → B is said to be quasi-Frobenius if it has a left adjoint L and right adjoint R such that L is similar to R. The notion of quasi-Frobenius functors is a symmetrized version of a left quasi-Frobenius pair of functors studied in [105]. C is called a quasi-Frobenius A-coring if the forgetful functor (−)A : MC → MA is quasi-Frobenius. Properties of such corings as well as quasi-Frobenius functors between comodule categories are studied in [71, 72].

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Frobenius properties of functors associated to morphisms of corings (or Frobenius coring extensions) are studied in [111, 203]. 5.3. Galois corings (comodules) and generalized descent Galois corings or Galois comodules are possibly the most studied class of corings in recent years. Galois corings can be understood as an approach to generalized Galois theory with its classical origins in ﬁeld extensions (see [198] for a review of the classical Galois theory in the context of comodules), which is still not quite well-understood from the categorical point of view developed in [35]. Furthermore, Galois corings lead to a reformulation of noncommutative descent theory [55]. Probably the strongest motivation of the recent interest in Galois comodules comes from noncommutative geometry; see Sections 6.2 and 6.3. 5.19. Galois comodules. Let C be an A-coring and (M, M ) a right C-comodule. ψN as Set S = End C (M). For any right A-module N, deﬁne a right C-comodule map a composition

ψN : HomA (M, N)⊗S M → N ⊗A C, f ⊗S m → f(m(0) )⊗A m(1) . (M, M ) is called a Galois comodule if and only if, for any right A-module N, the map ψN is an isomorphism of right C-comodules [199]. If M is ﬁnitely generated and projective as a right A-module, then HomA (M, N) N ⊗A M ∗ , where M ∗ = HomA (M, A). Thus, in this case, (M, M ) is a Galois comodule if and only if the canonical A-coring morphism ξ ⊗ m → ξ(m(0) )m(1) , canM : M ∗ ⊗S M → C, where M ∗ ⊗S M is the comatrix coring (cf. 3.11), is an isomorphism. A Galois comodule (M, M ) that is ﬁnitely generated and projective as an A-module is called a ﬁnite Galois comodule or the pair (C, M) is called a Galois coring [92]. The bijectivity of canM is referred to as a ﬁnite Galois condition. Furthermore, identifying M ∗ with HomC (M, C) by 4.10, one ﬁnds that the ﬁnite Galois condition is satisﬁed if and only if the evaluation map ψC : HomC (M, C) ⊗S M → C, f ⊗ m → f(m), is an isomorphism of right C-comodules. Note that the evaluation map is a counit of the coinvariants adjunction 4.11. In this chapter, we focus on ﬁnite Galois comodules, as these have been more thoroughly studied so far. For more information on general Galois comodules, the reader is referred to [199]. 5.20. A as a Galois comodule. The case in which A itself is a Galois C-comodule is of prime interest. As explained in 4.12, A is a right C-comodule if and only if there exists a group-like element g. In this case, the coinvariants S = End C (A) coincide with the subalgebra Ag = {s ∈ A | sg = gs} ⊆ A; see 4.14. Obviously, A is a ﬁnitely generated projective right A-module, A∗ A, and A ⊗S A is the Sweedler A-coring; see 3.10 and 3.11. The canonical map comes out as canA : A ⊗S A → C,

a ⊗ a → aga .

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Thus, A is a ﬁnite Galois comodule if and only if C is a Galois coring with respect to g, a notion introduced in [37]. 5.21. Galois connections associated to a ﬁnite Galois comodule. [79] Let M be a (B, A)-bimodule that is ﬁnitely generated and projective as a right A-module, and let C = M ∗ ⊗B M be the associated comatrix coring. Set T = End A (M) and consider the set Subext(T | B) of all algebras S such that B ⊆ S ⊆ T . Subext(T | B) is a partially ordered set with the order given by inclusion. Next, consider the set Coideal(C) of all coideals of C, cf. 3.4, and order it by K ≤ K ⇔ K ⊆ K. For any coideal K ∈ Coideal(C), view M as a right C/K-comodule with the coaction M : M → M ⊗A C/K, m → (M ⊗A p)(e ⊗A m), where e ∈ M ⊗A M ∗ is a dual basis element and p : C → C/K is the deﬁning epimorphism of K. Given any S ∈ Subext(T | B), deﬁne a coideal K(S) ∈ Coideal(C) as the kernel of the canonical map M ∗ ⊗B M → M ∗ ⊗S M. This deﬁnes an order-reversing operation K : Subext(T | B) → Coideal(C). Starting with a coideal K ∈ Coideal(C), deﬁne S(K) = End C/K (P) ⊆ T . In this way, one obtains an order-reversing operation S : Coideal(C) → Subext(T | B). Then, for all K ∈ Coideal(C) and S ∈ Subext(T | B), S (K (S)) ⊇ S,

K (S (K)) ⊆ K,

that is, the pair (K, S) is a Galois connection between the partially ordered sets Subext(T | B) and Coideal(C). Furthermore, S (K (S)) = S, if and only if ∗ End M ⊗S M (M) = S, and K (S (K)) = K, if and only if (M, M ) is a ﬁnite Galois C/K-comodule. 5.22. The ﬁnite Galois comodule structure theorem. This theorem describes properties of the M-coinvariants functor associated to a ﬁnite Galois comodule. It arose as a generalization of the structure theorem for Hopf–Galois extensions [167, Theorem 3.7] and was ﬁrst stated in [92, Theorem 3.2]. The present formulation includes further additions from [40, Theorem 2.1], [53, 18.27], [59]. Theorem. Let C be an A-coring and M be a right C-comodule that is ﬁnitely generated and projective as a right A-module. Set S = End C (M), and denote the M-coinvariants functor HomC (M, −) : MC → MS by G. (1) The following statements are equivalent: (a) M is a ﬁnite Galois comodule that is ﬂat as a left S-module. (b) C is a ﬂat left A-module and M is a generator in MC . (c) C is a ﬂat left A-module and G is fully faithful, that is, for any N ∈ MC , the counit of the coinvariants adjunction ψN : HomC (M, N) ⊗S M → N is an isomorphism of right C-comodules. (d) M is a ﬂat left S-module and G is fully faithful.

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277

(2) The following statements are equivalent: (a) M is a ﬁnite Galois comodule that is faithfully ﬂat as a left S-module. (b) C is a ﬂat left A-module and M is a projective generator in MC . (c) C is a ﬂat left A-module and G is an equivalence with the inverse −⊗S M : MS → MC . (d) M is a ﬂat left S-module and G is an equivalence. Part (1) is often referred to as the weak structure theorem, while part (2) is known as the strong structure theorem. It is worth pointing out that equivalences (b) ⇔ (c) ⇔ (d) are standard properties of Grothendieck categories and only (a) is special for comodules. 5.23. Semisimple and simple ﬁnite Galois comodules. A right C-comodule (M, M ) is called a semisimple comodule if, for any right C-comodule (N, N ), every right C-colinear monomorphism N → M is a section (of C-comodules). If any such monomorphism is an isomorphism, then (M, M ) is called a simple comodule. In case C is ﬂat as a left A-module, and hence MC is an Abelian category, (M, M ) is simple if and only if every monomorphism N → M is either 0 or an isomorphism, and it is semisimple if it is a direct sum of simple comodules. Following [199, 3.5], if C is a ﬂat left A-module, then (M, M ) is a semisimple comodule if and only if, for any set , the endomorphism ring of the direct sum comodule, End C (M () ), is a von Neumann regular ring and for any right C-comodule (N, N ), the counit of the M-coinvariants adjunction ψN : HomC (M, N) ⊗S M → N,

f ⊗ m → f(m),

is injective. Here, as before, S = End C (M). If, in addition, M is a ﬁnitely generated and projective right A-module, then (M, M ) is semisimple if and only if S is a (right) semisimple ring and, for all (N, N ), ψN is injective. Finally, by [40, Theorem 3.1], if C is ﬂat as a left A-module, and (M, M ) is ﬁnitely generated and projective as a right A-module, then (M, M ) is a simple comodule if and only if S is a division ring, and the counit of the M-coinvariants adjunction ψN is injective for any right C-comodule (N, N ). Therefore, every ﬁnite Galois comodule, whose endomorphism ring, is a division ring is a simple comodule. A (semi)simple comodule (M, M ) that is ﬁnitely generated and projective as a right A-module is a ﬁnite Galois comodule if and only if the evaluation map ψC is surjective and if and only if the canonical map canM is surjective. The deﬁnitions of a simple and semisimple comodules we use here in general differ from the deﬁnitions in [96] but coincide provided C is a ﬂat left A-module. 5.24. Simple cosemisimple corings and the Jacobson–Bourbaki correspondence. An A-coring is said to be cosemisimple if (C, C ) is a semisimple right C-comodule. This is a left–right symmetric notion [i.e. C is cosemisimple if and only if (C, C ) is a semisimple left C-comodule]. If C is cosemisimple, then C is projective (hence also ﬂat) as a left and right A-module; see [53, 19.14]. C is said to be simple if (C, C , C ) is a simple C-bicomodule.

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Cosemisimple corings were ﬁrst studied in [96]. In both [53, 96], they are termed semisimple corings. The term cosemisimple was introduced in [92] to be in-line with the accepted coalgebra terminology. Acoring C is simple and cosemisimple if and only if there is a ﬁnite Galois comodule (M, M ) such that the coinvariants algebra End C (M) is a division algebra. Furthermore, C is a simple and cosemisimple A-coring if and only if C is (isomorphic to) a comatrix coring associated to a (B, A)-bimodule P, where B is a division algebra. If B is another division algebra and P is a (B , A)-bimodule such that C is isomorphic to the comatrix coring associated to P , then P and P are isomorphic A-modules and B is conjugate to B by this isomorphism; see [92, Theorem 4.3]. Finally, any (nonzero) ﬁnitely generated right comodule over a simple cosemisimple coring is a ﬁnite Galois comodule, and its coinvariants algebra is simple Artinian [92, Proposition 4.2]. If C = M ∗ ⊗B M is a comatrix coring, then the Galois connection described in 5.21 restricts to mutually inverse maps on sets of simple Artinian subrings S of End A (M) [i.e. subextensions B ⊆ S ⊆ End A (M), where S is a simple Artinian algebra] and coideals K in C such that C/K is a simple cosemisimple A-coring [79, Theorem 2.4]. In case of an algebra map B → A, where A is a division algebra, setting M = A one obtains Sweedler’s predual version of the Jacobson– Bourbaki correspondence [178, Theorem 2.1]. It is worth noting here that some coringtype ideas appear already in Hochschild’s formulation of the Jacobson–Bourbaki theorem [110]. 5.25. Principal Galois comodules. Let (M, M ) be a right C-comodule; set S = End C (M) and consider any algebra map B → S. (M, M ) is called a B-relative principal Galois comodule if it is a ﬁnite Galois comodule that is B-relatively projective as a left S-module. Explicitly, for (M, M ) to be a B-relative principal Galois comodule, it is required that the canonical map canM : M ∗ ⊗S M → C,

ξ ⊗S m →

ξ(m(0) )m(1) ,

be bijective (the Galois condition) and that the S-multiplication map S ⊗B M → M,

s ⊗B m → s(m),

have a left S-linear section (the relative projectivity). (M, M ) is said to be a principal Galois comodule if it is Galois and projective as a left S-module. 5.26. Structure theorem for principal Galois comodules. Let C be an A-coring and (M, M ) a right C-comodule that is ﬁnitely generated and projective as a right A-module. View M ∗ = HomA (M, A) as a left C-comodule as in 4.6. Let S = End C (M), and let B → S be an algebra map such that S ⊗B M → C Hom(M ∗ , M ∗ ⊗B M),

s ⊗B m → [ξ → ξ ◦ s ⊗B m],

is an isomorphism of left S-modules. Then, the following statements are equivalent:

Comodules and Corings

(a) The map M : M ∗ ⊗B M → C, can

ξ ⊗B m →

279

ξ(m(0) )m(1) ,

is a split epimorphism of left C-comodules. (b) (M, M ) is a B-relative principal Galois comodule. Proof. The proof of implication (b) ⇒ (a) is straightforward; thus, we only comment on the proof (a) ⇒ (b). By assumption (a), the coring C is a direct summand of M ∗ ⊗B M as a left C-comodule; hence, in view of the hypothesis; M A Hom(M ∗ , A) C Hom(M ∗ , C) is a direct summand of a left S-module S ⊗ M, that is, M is a BB relatively projective left S-module. The evaluation map ϕM : M ∗ ⊗S C Hom(M ∗ , M ∗ ⊗B M) → M ∗ ⊗B M factorizes through the isomorphism S ⊗B M → C Hom(M ∗ , M ∗ ⊗B M) tensored with M ∗ and ϕM is an through the obvious isomorphism M ∗ ⊗S S ⊗B M → M ∗ ⊗B M. Hence isomorphism. Consider the following diagram, which is commutative in all possible directions, ∗ , M ∗ ⊗ M) M ∗ ⊗S C Hom(M B O

ϕM

/ M ∗ ⊗B M O

M) M ∗ ⊗S C Hom(M ∗ ,can

M ∗ ⊗S C Hom(M ∗ , C)

ϕC

M can

/C

where ϕC is the evaluation map. The upward pointing arrows are sections of M ∗ ⊗S C Hom(M ∗ , can M ) and can M , respectively. Since ϕM is an isomorphism, the map ϕC is bijective. The identiﬁcations C Hom(M ∗ , C) M and M ∗ HomC (M, C) lead to a k-linear isomorphism M ∗ ⊗S C Hom(M ∗ , C) HomC (M, C) ⊗S M. In view of this isomorphism, the fact that ϕC is bijective implies that the counit of adjunction ψC is bijective. Thus, M is a ﬁnite Galois right C-comodule. The above theorem is essentially the same as [40, Theorem 4.4]. The latter is formulated for algebras over a ﬁeld, that is, for the case when B = k is a ﬁeld, and the isomorphism S ⊗B M C Hom(M ∗ , M ∗ ⊗B M) is proved rather than assumed. The case when B = k is a commutative ring is described in [52, Theorem 2.1]. The case of a general B with a list of sufﬁcient conditions for the isomorphism S ⊗B M C Hom(M ∗ , M ∗ ⊗ M) is studied in [199, Theorem 5.9] using the correspondence B between Galois and ad-static comodules. The formulation given above follows that of [10, Theorem 5.1], with left–right conventions interchanged. Note that if M is a projective left B-module, then there is the required isomorphism S ⊗B M C Hom(M ∗ , M ∗ ⊗B M), and the resulting ﬁnite Galois module in part (b) is principal (not only B-relative principal). 5.27. Comonadicity and generalized descent. Categorical descent theory (cf. [138]) considers the following situation. With a monad T on a category B, one

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can associate the following (commutative, up to a natural isomorphism) triangle of categories and functors K / (BT )G B _??? ???? L yy ???? yy y ???? R ? |yyy UG . BT

Here BT is the category of T -algebras (or T -modules), R and UG are forgetful functors, L is a left adjoint of R (the induction functor), and G is the comonad LR. The functor K is the comparison functor deﬁned by K(B) = (L(B), L(ηB )), where η is the unit of the adjunction (L, R). The category (BT )G is called the category of descent data. In descent theory, one is interested in studying when K is a full and faithful functor (one then says that T is of descent type) and when it is an equivalence (one then says that T is of effective descent type). This is a categorical formulation of the descent for an algebra map, discussed in 4.20. To see this, take an algebra map B → A, take B to be the category of right B-modules and T to be the monad (−⊗B A, −⊗B μA , −⊗B 1A ). Then, BT is the category of right A-modules, G is the comonad on MA associated to the Sweedler coring A ⊗B A, and (BT )G is indeed the category of descent data Desc(A | B). Categorical descent theory ﬁts in a (yet) more general framework. Given categories A and B, a comonad G on A, a diagram of categories and functors K / AG B _??? } ?????L }} ???? }} R ??? ~}}} UG , A

in which L is the left adjoint of R, UG is the forgetful functor, and L = UG K, is called a G-comonadic triangle on A and B. We denote it by (L R, K)G . Start with an adjoint pair of functors (L, R), L : B → A, and write ψ and η for the counit and the unit, respectively. Then (LR, L(ηR ), ψ) is a comonad on A. By an argument dual to [88, Theorem II.1.1] (cf. [102, Theorem 2.2]), one can show that G-comonadic triangles (L R, K)G are in bijective correspondence with comonad morphisms ϕ : LR → G. Explicitly, given such ϕ, deﬁne β = ϕL ◦ L(η), and then, the functor K is constructed by setting K(B) = (L(B), βB ), for all objects B of B. Conversely, given a G-comonadic triangle (L R, K)G , the natural transformation ϕ is deﬁned as the composition ϕ = G(ψ) ◦ βR , where β : L → GL is the natural transformation deﬁned on objects B of B by the structure maps of K(B). In this general situation, one can study when the functor K is fully faithful or an equivalence. A complete solution to this problem, which is partly based on Beck’s theorem [17], can be found in [102, Theorems 2.6 and 2.7]. As explained in [102], the fact that the functor K in a comonadic triangle (L R, K)G is an equivalence is closely related to the fact that ϕ is an isomorphism. More precisely, given (L R, K)G and the corresponding ϕ, assume that, for all

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G-coalgebras, (M, M ), there exists in B an equalizer of R(M ) and R(ϕM ) ◦ ηR(M) . Then K is an equivalence if and only if the functor L preserves all such equalizers and the map ϕ is an isomorphism of comonads. The relevance of comonadic triangles to Galois properties of comodules of an A-coring C can be explained as follows. Take A = MA and G = − ⊗A C, the comonad associated to C. Thus AG = MC . Fix a right C-comodule (M, M ) and let S = End C (M). Let B = MS . Since the homomorphism functor is the right adjoint of the tensor functor, this can be recast as the following comonadic triangle K / MC MS D aDDDD −⊗ M z DDDD S zz DDDD zz(−) z D D A z HomA (M,−) D ! }z , MA

where K(N) = (N ⊗S M, N ⊗S M ). The natural transformation ϕ associated to this triangle comes out as ψ as deﬁned in 5.19. Thus, the (ﬁnite) Galois property of (M, M ) is a necessary condition for the effectiveness of generalized descent associated with (M, M ) (note that M is a ﬁnitely generated and projective as a right A-module by [187, Corollary 3.7]). 5.28. Strong structure theorem over nonunital rings [102, Theorem 3.5]. Let C be an A-coring, B be a nonunital ﬁrm algebra. To any (B, C)-bicomodule (M, M ) (M is ﬁrm as a left B-module), one can associate the following comonadic triangle K / MC MB D aDDDD −⊗ M z DDDD B zz DDDD zz(−) z D D A HomA (M,−)⊗B B D ! }zz , MA

where K(N) = (N ⊗B M, N ⊗B M ). Applying 5.27 to this triangle, one concludes that K is an equivalence of categories if and only if M is a faithfully ﬂat left B-module and, for all right A-modules N, the map HomA (M, N) ⊗B B ⊗B M → N ⊗A C, f ⊗B b ⊗B m → f(bm(0) ) ⊗A m(1) , is an isomorphism. 5.4. Morita theory and corings There are several Morita contexts, which can be associated to corings and comodules. They can be used to formulate structure theorems for Galois comodules or to study equivalences between module categories. In this section, we describe Morita contexts associated to corings. 5.29. Morita contexts. A Morita context or a set of pre-equivalence data is a sextuple (A, B, N, M, σ, τ), where

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(i) A and B are algebras; (ii) M is a (B, A)-bimodule and N is an (A, B)-bimodule; (iii) τ : M ⊗A N → B is a B-bimodule map and σ : N ⊗B M → A is an A-bimodule map rendering the following diagrams commutative N ⊗B M ⊗A N

σ⊗A N

/ A ⊗A N

N⊗B τ

N ⊗B B

M ⊗A N ⊗B M

/ B ⊗B M

M⊗A σ

M ⊗A A

/N,

τ⊗B M

/M

.

A Morita context (A, B, N, M, σ, τ) is said to be strict provided both σ and τ are isomorphisms. Any strict Morita context induces and arises from an equivalence of categories of modules MA and MB (in which case one says that the algebras A and B are Morita equivalent). For a detailed discussion of Morita contexts or pre-equivalence data, the reader is referred to [14, Chapter II.3]. 5.30. Morita context as a two-object category [27, Remark 3.2(1)]. A k-linear category A with two objects a and b gives rise to a Morita context as follows. The composition of morphisms makes k-modules M = A(a, b) and N = A(b, a) bimodules for k-algebras (with product given by composition) A = A(a, a) and B = A(b, b). Furthermore, the composition of maps deﬁnes bimodule maps τ : A(a, b) ⊗A A(b, a) → B,

σ : A(b, a) ⊗B A(a, b) → A.

One easily checks that (A, B, A(b, a), A(a, b), σ, τ) is a Morita context. Clearly, there is a category of this kind behind any Morita context. 5.31. Morita context associated to a comodule I [59, Section 4]. The ﬁrst Morita context connects the endomorphism algebra of a comodule with the right dual ring of a coring. Take a right comodule (M, M ) over an A-coring C. (i) Let S = End C (M) be the endomorphism algebra of M and B be the opposite algebra to the right dual ring C ∗ ; see 3.25. (ii) Let Q = HomC (C, M) and M ∗ = HomA (M, A). Then Q is an (S, B)-bimodule with the multiplications given by, for all s ∈ S, b ∈ B, q ∈ Q, and c ∈ C, (qb)(c) =

q(b(c(1) )c(2) ),

sq = s ◦ q,

and M ∗ is a (B, S )-bimodule with the multiplications given by, for all s ∈ S, b ∈ B, n ∈ M ∗ , and m ∈ M, (bn)(m) =

b(n(m(0) )m(1) ),

(iii) Deﬁne an S-bimodule map σ : Q ⊗B M ∗ → S,

ns = n ◦ s.

q ⊗B n → m → q(n(m(0) )m(1) ) ,

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and a B-bimodule map τ : M ∗ ⊗S Q → B,

n ⊗S q → n ◦ q.

Then M∗ = (S, B, Q, M ∗ , σ, τ) is a Morita context. In view of 4.10, B End C (M) and M ∗ HomC (M, C). Hence M∗ is a context associated to the full subcategory of MC consisting of the two objects (M, M ) and (C, C ); cf. [189, Proposition 5.10]. A similar context can be constructed for a left C-comodule. 5.32. Morita context associated to a comodule II [33, Section 2]. This Morita context connects the endomorphism algebra of a comodule with the left dual ring of a coring. Take a right comodule (M, M ) of an A-coring C. (i) Let S = End C (M) be the endomorphism algebra of M and B be the opposite algebra to the left dual ring ∗ C, that is, bb = b ∗l b; see 3.25. (ii) Let q(m(0) )(c)m(1) Q = q ∈ HomA (M, B) | ∀m ∈ M, c ∈ C, = c(1) q(m)(c(2) ) . Then Q is a (B, S)-bimodule with the multiplications (bqs)(m) = bq(s(m)), for all s ∈ S, b ∈ B, q ∈ Q, and m ∈ M. M is an (S, B)-bimodule with the multiplications smb = s(m(0) b(m(1) )); cf. 4.33. (iii) Deﬁne an S-bimodule map σ : M ⊗B Q → S,

m ⊗B q → [m → mq(m )],

and a B-bimodule map τ : Q ⊗S M → B,

q ⊗S m → q(m).

Then M = (S, B, M, Q, σ, τ) is a Morita context. Recall from 4.6 that if M is ﬁnitely generated projective as a right A-module, ∗ then (M ∗ , M ) is a left C-comodule. In this case, the Morita context M is the same ∗ as the Morita context M∗ in 5.31 associated to the left comodule (M ∗ , M ); see [33, Remark 2.3]. In the case, C is a locally projective left A-module, the bimodule Q is isomorphic to the B-dual module HomB (M, B); see [33, Remark 2.2]. In this case, M is a Morita context associated to the full subcategory of MB consisting of the two objects M and B (cf. 5.30). The context M controls the behavior of the functor − ⊗S M : MS → MC . If the map σ in M is surjective, then − ⊗S M is a fully faithful functor [33, Theorem 2.6]. If C is ﬁnitely generated and projective as a left A-module, then M is strict if and only if (M, M ) is a ﬁnite Galois comodule and M is faithfully ﬂat as a left S-module [59, Theorem 4.12], [33, Proposition 2.7] (and hence − ⊗S M is an equivalence by the ﬁnite Galois comodule structure theorem 5.22). 5.33. Coring extensions [41]. In case of algebras, an algebra map is the same as an algebra extension. More explicitly, given two algebras A and B, an algebra extension

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or an algebra map B → A can be equivalently characterized as a k-linear functor F : MA → MB with the factorization property F

/ MB MA E EE y y EE yy E yyUB UA EE y " |y , Mk , where UA and UB are forgetful functors (cf. [152]). Extending this functorial characterization of algebra extensions, one says that a B-coring D is a (right) functorial coring extension of an A-coring C if there exists a k-linear functor F : MC → MD with the factorization property F / MD MC D DD z z DD zz D zz (−)A DD ! |zz (−)B MB MA E EE zz EE z z E zz UA EE " |zz UB , Mk ,

where (−)A , and (−)B are forgetful functors. If the above diagram can be completed commutatively by a functor MA → MB , then functorial coring extension is also known as a cofunctor (between internal cocategories); see [8, Section 4.2]. Following [41, Deﬁnition 2.1], D is called a (right) coring extension of C if there exists a right coaction κ : C → C ⊗B D such that (C, κ, C ) is a (C, D)-bicomodule. A coring extension is denoted by (D | C; κ). Every functorial coring extension gives rise to a coring extension (cf. [41, (2) ⇒ (1) Theorem 2.6]3 ). Given a functor F : MC → MD , the coaction κ is induced from the right regular coaction C , that is, κ = F(C ). Conversely, given a right coring extension (D | C; κ), the functor F : MC → MD can be deﬁned by F : (M, M ) → (M, F(M )), F(M ) : M

M

/ M2C C

M2C κ

/ M2C (C ⊗B D) M ⊗B D,

provided that, for all C-comodules M, the right B-module map M ⊗A C − M ⊗A C is D ⊗B D-pure. An extension (D | C; κ) satisfying this purity condition is called a pure (functorial) coring extension. For example, if D is a ﬂat left B-module, or D is a coring associated to an entwining structure as in 3.21, then any coring extension (D | C; κ) is pure (hence functorial). 3 Unfortunately, the proof of the claim (1) ⇒ (2) of Theorem 2.6 in [41] is incorrect without further assumptions discussed in this section.

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If B = A, then a coring map γ : C → D induces a coring extension (D | C; (C ⊗A γ) ◦ C ). Every coring extension (D | C; κ), in which C is a right D-comodule with its original A-multiplication, induces an A-coring morphism γ : C → D by γ = (εC ⊗A D) ◦ κ; see [28, Corrigendum, Lemma 1]. If D and C are corings associated to entwining structures (see 3.20 and 3.21), then extensions (D | C; κ) correspond to morphisms of entwining structures in the sense of [153]. Take an entwining structure (A, C)ψ and let C = A⊗C be the associated A-coring. Then C is a pure coring extension of C, (C | C; A ⊗ C ). 5.34. Morita context for a pure coring extension [33, Proposition 3.1]. Let C be an A-coring and D be a B-coring, and let (D | C; κ) be a pure coring extension. Take a (B, C)-bicomodule (M, M ), and set S = End C (M). Since M is left B-linear, there is an algebra map j : B → S; hence S is a B-bimodule. (i) View R = B HomB (D, S ) as an algebra with the convolution product, rr (d) = r(d (1) )r (d (2) ) and unit j ◦ εD , and set T to be the opposite of the endomorphism ring C End D (C) (i.e. the product in T is tt = t ◦ t). (ii) Let N = B HomD (D, M) and c(1) q(c(2) )(m) Q = q ∈ A HomB (C, M ∗ ) | ∀m ∈ M, c ∈ C, = q(c)(m(0) )m(1) , where M ∗ = HomA (M, A) is an (A, B)-bimodule by (afb)(m) = af(bm). Then N is an (R, T )-bimodule with multiplications, for all n ∈ N, r ∈ R, t ∈ T , and d ∈ D, (rn)(d) = r(d (1) )(n(d (2) )), (nt)(d) = n(d)(0) εC t n (d) (1) , and Q is a (T, R)-bimodule with multiplications, for all q ∈ N, r ∈ R, t ∈ T , and c ∈ C, qt(c) = q(c[0] )t(c[1] ), tq = q ◦ t, where c[0] ⊗A c[1] = κ(c). (iii) Deﬁne a T -bimodule map τ : Q ⊗R N → T, and an R-bimodule map σ : N ⊗T Q → R,

q ⊗R n → c → c(1) q(c(2) [0] ) n c(2) [1] , n ⊗T q → d → n(d)(0) q n (d) (1) (−) .

Then BV = (R, T, N, Q, σ, τ) is a Morita context. When interpreted as a category with two objects as in 5.30, the Morita context BV arises from a category whose objects are functors: the functor F : MC → MD , which

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T. Brzezi´nski

deﬁnes the pure coring extension (D | C; κ), and the functor HomC (M, −) ⊗B D : MC → MD . Morphisms in this category are natural transformations. 5.35. Morita contexts and structure theorems for Galois comodules [33, Theorems 3.6, 4.1, 4.6]. Let BV = (R, T, N, Q, σ, τ) be the Morita context constructed in 5.34. (1) The map τ is surjective if and only if M is a Galois comodule, and, as an (S, D)bicomodule, M is a direct summand of a ﬁnite number of copies of S ⊗B D. In particular, if τ is surjective, then the M-coinvariants functor HomC (M, −) : MC → MS is fully faithful. (2) The M-coinvariants functor is an equivalence, provided the context BV is strict, and there exists a ﬁnite number of elements ri ∈ B HomB (D, S ), di ∈ D, such that i ri (di ) = 1S . In this case, M is a ﬁnitely generated and projective right A-module (hence a ﬁnite Galois comodule, but not necessarily faithfully ﬂat as a left S-module). 5.36. Cleft bicomodules and structure theorems [33, Section 5]. Let C be an A-coring and D be a B-coring, and let (D | C; κ) be a pure coring extension. Take a (B, C)-bicomodule (M, M ) and consider the associated Morita context BV = (R, T, N, Q, σ, τ). If there exist q ∈ Q and n ∈ N such that τ(q ⊗R n) = C,

σ(n ⊗T q) = εD (−)1S ,

(M, M )

then is said to be a cleft bicomodule. Following [3, Deﬁnition 4.6], an entwining structure (A, C)ψ is said to be cleft if A is an entwined module, and there exists a convolution invertible right C-comodule map C → A. Set C = A ⊗ C to be the coring associated to (A, C)ψ . Then A is a cleft bicomodule, with respect to the coring extension (C | C; A ⊗ C ), if and only if (A, C)ψ is a cleft entwining structure. Any cleft bicomodule is a Galois comodule. Furthermore, the M-coinvariants functor of a cleft bicomodule (M, M ) is fully faithful, and it is an equivalence, provided there exists a ﬁnite number of elements ri ∈ B HomB (D, S), di ∈ D, such that i ri (di ) = 1S . 5.5. Bialgebroids A bialgebroid is a generalization of the notion of a bialgebra to the case of a noncommutative base ring. Thus, this is a coring with a compatible algebra structure. The structure of bialgebroids and Hopf algebroids is treated in full in [25]. Here, we only give the most basic deﬁnitions. 5.37. Ae -rings. Let Ae = A ⊗ Aop be the enveloping algebra of an algebra A. An algebra H is an Ae -ring if and only if there exist an algebra map s : A → H and an antialgebra map t : A → H such that s(a)t(b) = t(b)s(a), for all a, b ∈ A. A is called a base algebra, H a total algebra, s the source map and t the target map. To indicate explicitly the source and target maps, we write (H, s, t).

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5.38. The Takeuchi product [179, 181]. Take a pair of Ae -rings (U, sU , tU ) and (V, sV , tV ), view U as a right A-module via the left multiplication by the target map tU and view V as a left A-module via the left multiplication by the source map sV . A Takeuchi ×A -product is then deﬁned as

U ×A V :=

i

ui ⊗A vi ∈ M ⊗A N | ∀ b ∈ A,

i

ui tU (b) ⊗A vi =

i

ui ⊗A vi sV (b) .

The importance of the notion of the Takeuchi ×A -product is a direct consequence of the following observation ([179, Proposition 3.1], [181, Proposition 3.1]). Lemma. For any pair of Ae -rings (U, sU , tU ) and (V, sV , tV ), the Ae -bimodule U×A V is an Ae -ring with source a → sU (a)⊗A 1V and target a → 1U ⊗A tV (a), associative product ui ⊗A vi u˜ j ⊗A v˜ j = ui u˜ j ⊗A vi v˜ j , i

j

i,j

and unit 1U ⊗A 1V . 5.39. Bialgebroids [131,181]. Let (H, s, t) be an Ae -ring. View H as an A-bimodule by ah = s(a)h, ha = t(a)h, for all a ∈ A, h ∈ H. We say that (H, s, t, H , εH ) is a left A-bialgebroid if (1) (H, H , εH ) is an A-coring; (2) Im(H ) ⊆ H ×A H and the corestriction of H to H : H → H ×A H is an algebra map; (3) εH (1H ) = 1A , and, for all g, h ∈ H, εH (gh) = εH (gs(εH (h))) = εH (gt(εH (h))) . 5.40. The monoidal structure of the category of (co) modules [165]. Let (H, s, t) be an Ae -ring. Then H is an A-bialgebroid if and only if H M is a monoidal category such that the forgetful functor F : H M → A MA is strict monoidal. Furthermore, if H is an A-bialgebroid, the category H M of left comodules of the A-coring H is monoidal with the tensor product deﬁned by (M, M) ⊗ (N, N) = (M ⊗A N, M⊗A N), where, writing M(m) = m(−1) ⊗A m(0) , N(n) = n(−1) ⊗A n(0) , for the left H-coactions, M⊗A N : m ⊗A n → m(−1) n(−1) ⊗A m(0) ⊗A n(0) . The right A-multiplication on a left H-comodule (M, M) is given by ma = εH (m(−1) s(a))m(0) .

288

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This is a unique right A-module structure which makes H an A-bimodule and such that the image of the coaction M is contained in the Takeuchi product H ×A M; see [23, Section 2.2]. 6. Applications 6.1. Hopf modules Recent interest in the theory of corings has arisen from their connection with Hopftype modules, studied in Hopf algebra theory since the 1960s. This connection is provided through the observation that all such Hopf-type modules can be interpreted as entwined modules and hence as comodules of corings. In this section, we merely list the known types of Hopf modules and describe the corresponding entwining structures. More details can be found in [53, 68]. 6.1. Bialgebras and Hopf algebras. A bialgebra can be deﬁned as a special case of an A-bialgebroid by specifying A to be k. Explicitly, a bialgebra is a k-module B such that (B, μB , 1B ) is a k-algebra, (B, B , εB ) is a k-coalgebra, and both B and εB are algebra maps, that is, for all a, b ∈ B, εB (ab) = εB (a)εB (b), and B (ab) =

εB (1B ) = 1,

a(1) b(1) ⊗ a(2) b(2) ,

B (1B ) = 1B ⊗ 1B .

A bialgebra B is called a Hopf algebra if there exists a k-linear map S : B → B such that μB ◦ (B ⊗ S ) ◦ B = 1B ◦ εB = μB ◦ (S ⊗ B) ◦ B . The map S is called an antipode. For more details about bialgebras and Hopf algebras, the reader is referred, for example, to [75]. 6.2. Comodule algebras and module coalgebras. Let B be a bialgebra. An algebra A is called a right B-comodule algebra if there exists an algebra map A : A → A⊗B such that (A, A ) is a right B-comodule. The algebra map property of A means that, for all a, b ∈ A, a(0) b(0) ⊗ a(1) b(1) , A (1B ) = 1A ⊗ 1B . A (ab) = Dually, a coalgebra C is called a right B-module coalgebra if C is a right B-module and, for all b ∈ B and c ∈ C, C (cb) = c(1) b(1) ⊗ c(2) b(2) , εC (cb) = εC (c)εB (b). 6.3. Hopf modules. Let B be an algebra and a coalgebra. The k-linear map b(1) ⊗ b b(2) , ψ : B ⊗ B → B ⊗ B, b ⊗ b →

Comodules and Corings

289

entwines B with B if and only if B is a bialgebra. An entwined module corresponding to (B, B)ψ is a right B-comodule M such that for all m ∈ M and b ∈ B, M (mb) =

m(0) b(1) ⊗ m(1) b(2) .

Any such M is known as a Hopf module, a notion introduced and studied already in [129, 177]. Thus if B is a bialgebra, then C = B⊗B is a B-coring with coproduct C = B⊗B , counit εC = B ⊗ εB , and the B-bimodule structure bb c(1) ⊗ b c(2) , b(b ⊗ b )c = for b, b , b , c ∈ B. The category of right C-comodules is the same as the category of (right) B-Hopf modules. 6.4. Relative Hopf modules. Let B be a bialgebra and let A be an algebra together with a k-linear map A : A → A ⊗ B, a → a(0) ⊗ a(1) . Then the map ψ : B ⊗ A → A ⊗ B,

b ⊗ a →

a(0) ⊗ ba(1) ,

entwines A with B if and only if (A, A ) is a right B-comodule algebra (cf. [63, Lemma 4.1]). The corresponding entwined modules are right A-modules and right B-comodules (M, M ) such that, for all m ∈ M and a ∈ A, M (ma) = m(0) a(0) ⊗ m(1) a(1) . Such a module is known as a relative (B, A)-Hopf module, a notion introduced in [182] (cf. [83]). Thus if A is a right B-comodule algebra, then C = A ⊗ B is an A-coring with coproduct C = A ⊗ B , counit εC = A ⊗ εB , and the A-bimodule structure, for a, a , a ∈ A and b ∈ B, a (a ⊗ b)a = a aa (0) ⊗ ba (1) . The category of right C-comodules is the same as the category of relative (B, A)-Hopf modules. Dualizing the above construction, one entwines a bialgebra B and a right B-module coalgebra C by the map ψ : C ⊗ B → B ⊗ C, c ⊗ b → b(1) ⊗ cb(2) . The corresponding B-coring C = B ⊗ C has a coproduct C = B ⊗ C , counit εC = B ⊗ εC , and the (B, B)-bimodule structure, for b, b , b ∈ B, and c ∈ C, b (b ⊗ c)b = b bb (1) ⊗ cb (2) .

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Comodules over C are identiﬁed with right C-comodules right B-modules M such that the coaction is compatible with the action via M (mb) = m(0) b(1) ⊗ m(1) b(2) . Such an M is called a [B, C]-Hopf module [83]. 6.5. Doi–Koppinen Hopf modules. All the above examples of Hopf modules are special cases of Doi–Koppinen Hopf modules introduced in [84, 126]. In this case, one starts with a triple (B, A, C), where B is a bialgebra, A is a right B-comodule algebra, and C is a right B-module coalgebra. A is then entwined with C by the map ψ : C ⊗ A → A ⊗ C, c ⊗ a → a(0) ⊗ ca(1) , where A (a) = a(0) ⊗a(1) ∈ A⊗B is the B-coaction on A. Thus the corresponding A-coring C = A ⊗ C has the coproduct C = A ⊗ C , counit εC = A ⊗ εC , and the A-bimodule structure, for a, a , a ∈ A, and c ∈ C, a a a(0) ⊗ ca(1) . a (a ⊗ c)a = Right C-comodules are identiﬁed with right A-modules and right C-comodules M with the action-coaction compatibility, for all a ∈ A and m ∈ M, M (ma) = m(0) a(0) ⊗ m(1) a(1) . The triple (B, A, C) satisfying above conditions is known as a Doi–Koppinen datum. Doi–Koppinen data were considered in [126, Deﬁnition 2.1], where they were termed opposite smash data. This was the ﬁrst (recorded) step toward understanding Doi–Koppinen Hopf modules as representations of an algebraic structure (hence, toward viewing Hopf modules as comodules of a coring). Finally, the entwining structure associated to a Doi–Koppinen datum was ﬁrst introduced in [36]. The left dual algebra of the coring A ⊗ C is isomorphic to the Koppinen (opposite) smash product algebra [126]. Many properties and applications of Doi–Koppinen structures are studied in the monograph [68]. 6.6. Yetter–Drinfeld modules. Given a Hopf algebra H with antipode S, a (right– right) Yetter–Drinfeld module is a right H-module and a right H-comodule (M, M ) such that, for all m ∈ M and h ∈ H, m(0) h(2) ⊗ (Sh(1) )m(1) h(3) M (mh) = [158,201]. As explained in [66,67], Yetter–Drinfeld modules are an example of Doi– Koppinen Hopf modules, hence entwined modules. The entwining map is ψ : H ⊗ H → H ⊗ H, h ⊗ h → h(2) ⊗ (Sh(1) )h h(3) .

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Thus Yetter–Drinfeld modules are right comodules over the H-coring C = H ⊗ H with coproduct C = H ⊗ H , counit εC = H ⊗ εH , and the H-bimodule structure, for all h, h , h , k ∈ H, k(h ⊗ h )h = kh h(2) ⊗ (Sh(1) )h h(3) . 6.7. Anti-Yetter–Drinfeld modules. This most recent addition to Hopf modules appeared in [107, 108, 113] as coefﬁcients for Hopf-cyclic homology [77]. Let H be a Hopf algebra with a bijective antipode S. An anti-Yetter–Drinfeld module is a right H-module and H-comodule M with the action-coaction compatibility condition, for all a ∈ H, m ∈ M, m(0) a(2) ⊗ S −1 (a(1) )m(1) a(3) . M (ma) = As explicitly explained in [43], anti-Yetter–Drinfeld modules are entwined modules for the entwining map ψ : H ⊗ H → H ⊗ H given by ψ(c ⊗ a) = a(2) ⊗ S −1 (a(1) )ca(3) , for all a, c ∈ H. The existence of entwining means that C = H ⊗ H is an H-coring with coproduct C = H ⊗ H , counit εC = H ⊗ εH , and the H-bimodule structure, for all h, h , h , k ∈ H, k(h ⊗ h )h = kh h(2) ⊗ S −1 (h(1) )h h(3) . Anti-Yetter–Drinfeld modules are right comodules over the coring C. In the deﬁnition of both Yetter–Drinfeld and anti-Yetter–Drinfeld modules, the important property is that the antipode is an antibialgebra map and the identity map is a bialgebra map. Following this observation, the notion of (α, β)-equivariant C-comodules was introduced in [150]. The deﬁning data comprise a bialgebra A, A-bimodule coalgebra C, a bialgebra map α : A → A, and an antibialgebra map β : A → A (i.e. β is both an antialgebra and anticoalgebra map). All these data give rise to an entwining map ψ : C ⊗ A → A ⊗ C, ψ(c ⊗ a) = a(2) ⊗ β(a(1) )cα(a(3) ), and thus to an A-coring C = A⊗C with coproduct C = A⊗C , counit εC = A⊗εC , and the A-multiplication a (a ⊗ c)a = a a a(2) ⊗ β(a(1) )cα(a(3) ). The (α, β)-equivariant C-comodules are simply comodules over C. 6.8. Weak Doi–Koppinen modules. The notion of a weak bialgebra was introduced in [32, 145] (cf. [30]), by weakening the axioms of a bialgebra. A weak k-bialgebra H is a k-module with a k-algebra structure (μH , 1H ) and a k-coalgebra structure

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(H , εH ) such that H is a multiplicative map and (H ⊗ H ) ◦ H (1H ) = 1H (1) ⊗ 1H (2) 1H (1 ) ⊗ 1H (2 ) = 1H (1) ⊗ 1H (1 ) 1H (2) ⊗ 1H (2 ) , εH (hkl) = εH (hk(1) )εH (k(2) l) = εH (hk(2) )εH (k(1) l) for all h, k, l ∈ H. Here H (1H ) = 1H (1) ⊗ 1H (2) = 1H (1 ) ⊗ 1H (2 ) . Given a weak bialgebra H, a right H-comodule algebra is deﬁned in [22, Definition 2.1] (cf. [56, Proposition 4.12]) as a k-algebra A with a right H coaction A : A → A ⊗ H such that, for all a, b ∈ A, A (ab) = A (a)A (b), a(0) ⊗εH (1H (1) a(1) )1H (2) = 1A(0) a⊗1A(1) . Dually, a right H-module coalgebra is deﬁned as a coalgebra (C, C , εC ) and a right H-module such that, for all h, k ∈ H, and c ∈ C, C (ch) = c(1) h(1) ⊗ c(2) h(2) , εC (chk) = εC (ch(2) )ε(h(1) k). Given a right H-comodule algebra A and a right H-module coalgebra C, one deﬁnes a k-linear map ψ : C ⊗ A → A ⊗ C, c ⊗ a → a(0) ⊗ ca(1) . The triple (A, C, ψ) is a right-right weak entwining structure (cf. [56, Theorem 4.14]), and hence there is an associated A-coring C as described in 3.23. The comodules over this coring are identiﬁed with right A-modules and right C-comodules M that satisfy the action–coaction compatibility condition m(0) a(0) ⊗ m(1) a(1) . M (ma) = Such modules were ﬁrst introduced and studied in [22] and are termed weak Doi– Koppinen Hopf modules. A triple comprising a weak bialgebra H, a right H-comodule algebra A, and a right H-module coalgebra C is known as a weak Doi–Koppinen datum. 6.9. Hopf modules as comodules of a coring. Once it is realized that all known types of Hopf modules can be seen as comodules of speciﬁc corings, the general results of the coring theory can be used to gain information about Hopf modules. In all cases, the coring arises from an entwining structure (A, C)ψ hence is of the type C = A ⊗ C. The deﬁning adjunction (−)A , − ⊗A C can be identiﬁed with the adjunction C (ψ) → MA , (−)A : MA

C − ⊗ C : MA → MA (ψ),

where, for any right A-module V , V ⊗ C is a right C-comodule by V ⊗ C and a right A-module by (v ⊗ c)a = vψ(c ⊗ a). Using knowledge about the category of comodules over a coring, one thus derives conditions for a category of speciﬁc Hopf

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modules to be Grothendieck or to be a full subcategory of the category of modules over the corresponding smash (i.e. twisted convolution) algebra. Furthermore, properties of the functors − ⊗ C and (−)A such as separability or the Frobenius property are derived from properties of the deﬁning adjunction of the coring C = A ⊗ C and thus correspond to C being a cosplit, coseparable, or Frobenius coring. More geometric properties such as the interpretation of Hopf modules as modules with ﬂat connections discussed in [121] are a direct consequence of the interpretation of comodules over a coring as ﬂat connections. Similar comments apply to weak Doi–Koppinen Hopf modules. There are, however, limitations as to this use of corings in description of Hopf modules. First, every class of Hopf modules has its speciﬁc properties, which are not captured by the deﬁnition of a coring. For example, the category of Yetter–Drinfeld modules is a braided monoidal category. This property and its consequences cannot be derived from the general properties of corings (without specifying the coring precisely). Second, the notion of an entwining structure is self-dual in the sense that replacing A by C, μA by C , 1A by εC , and reversing all the arrows in the Diagram 3.20 produces exactly the same conditions. Using this self-duality, one can translate properties of functors (−)A , − ⊗ C into corresponding properties of functors C (−)C : MA (ψ) → MC ,

C − ⊗ A : MC → MA (ψ),

where (−)C is the forgetful functor and, for any right C-comodule V , the C-coaction for V ⊗ A is v ⊗ c → v(0) ψ(v(1) ⊗ a). Obviously, there can be no functor such as (−)C for a general A-coring C. Thus to have a more unifying point of view on these properties of Hopf modules, one needs to study algebras in the category of bicomodules over a coalgebra C or C-rings; see Section 7.2. 6.2. Noncommutative differential geometry We have already seen in 3.16 that corings with a group-like element give rise to (semifree) differential graded algebras. Furthermore, as explained in 4.23, comodules over a coring with a group-like element are the same as modules with a ﬂat connection (with respect to the induced differential graded algebra). This is one of the situations, where corings relate to noncommutative geometry. The second is the appearance of comodules over a coring (of the anti-Yetter–Drinfeld type) as coefﬁcients for the Hopf-cyclic homology; see 6.7. Another one is the geometry of noncommutative principal bundles. As this is not a text about geometry or addressed to geometers, we recall brieﬂy the basic notions, and we provide an algebraic description of a geometric situation that makes it possible to relate noncommutative principal bundles to Galois type extensions. For a detailed and easy accessible noncommutative description of classical bundles, we refer to [15]. 6.10. Free actions and principal bundles. Let X be a compact Hausdorff space and G a compact group. The action of G on X is said to be free if the property xg = x for any x ∈ X implies that g = u, the group identity.

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G : X × G → X × X, (x, g) → (x, xg). The freeness of the Consider the map F X

G is injective. Write Y = X/G G-action on X means, equivalently, that the map F X and let π : X → Y , x → xG, be the canonical surjection. Let X ×Y X := {(x, y) ∈

G ; so, the action of G on X is free X × X | π(x) = π(y)}. Clearly, X ×Y X = Im F X if and only if F˜ XG : X × G → X ×Y X,

(x, g) → (x, xg),

is bijective. Given a free action of G on X, the function τˆ : X ⊗Y X → G, (x, xg) → g, is called a translation function. X is called a G-principal bundle over Y if and only if τˆ is continuous. X is called the total space, Y is called the base space, and G is called the structure group or the ﬁber. 6.11. Algebras of functions on X, G, Y and X ×Y X. Consider now algebras of (complex, polynomial – thus we assume that all spaces are embedded in Cn ) functions A = O(X) and H = O(G). Use the identiﬁcation O(G × G) O(G) ⊗ O(G) to see that H is a Hopf algebra with the coproduct H (f )(g, h) = f(gh) and counit εH (f ) = f(u), where u is the neutral element of G. The action of G on X translates into the coaction of H on A A : A → A ⊗ H, A (a)(x, g) = a(xg). Obviously, since both H and A are commutative algebras, A is an algebra map; hence A is a right H-comodule algebra. Let B = O(Y ), the algebra of (polynomial) functions on Y = X/G. We can view B as a subalgebra of A via the pullback of π : X → Y , that is, via the function π∗ : B → A, b → b ◦ π. The map π∗ is injective, since b = b means that there exists at least one coset xG such that b(xG) = b (xG). But xG = π(x); so we conclude that π∗ (b)(x) = π∗ (b )(x). Furthermore, a ∈ π∗ (B) if and only if a(xg) = a(x), for all x ∈ X, g ∈ G. This is the same as A (a)(x, g) = (a ⊗ 1)(x, g), for all x ∈ X, g ∈ G, where 1 : G → k is the unit function 1(g) = 1 [the unit of the Hopf algebra H = O(G)]. Thus, we can identify B with the coinvariants: B = AcoH := {a ∈ A | A (a) = a ⊗ 1H }. Since A is an algebra map, AcoH can be described equivalently as AcoH = {b ∈ A | ∀a ∈ A, A (ba) = bA (a)}. B is a subalgebra of A (via the map π∗ ), hence it acts on A through the inclusion map, (bab )(x) = b(π(x))a(x)b (π(x)). We can identify O(X ×Y X) with O(X) ⊗O(Y) O(X) = A ⊗B A, via the map θ(a ⊗B a )(x, y) = a(x)a (y), with π(x) = π(y). 6.12. Principal bundles as Hopf–Galois extensions. With X, G, Y as before, the action of G on X is free if and only if the map ∗

FXG : O(X ×Y X) → O(X × G),

f → f ◦ FXG , ∗

is bijective. In view of the identiﬁcations in 6.11, FXG is the canonical map canA : a ⊗B a → [(x, g) → a(x)a (xg)] = aA (a ).

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Thus the action of G on X is free if and only if the map canA is bijective, that is, the extension of algebras B ⊆ A is a Hopf-Galois extension by H. Here A = O(X), H = O(G), and B = O(X/G). To deal with the topological aspects, one needs to consider C∗ -algebras of continuous functions (in which, by the Stone-Weierstrass theorem, the polynomial algebras are dense subalgebras) and complete tensor products. 6.13. Coalgebra-Galois extensions. The algebraic formulation of the notion of a principal bundle leads to the following deﬁnition. Let C be a k-coalgebra and A a k-algebra and a right C-comodule with coaction A : A → A ⊗ C. Let B be the subalgebra of coinvariants of A, B := {b ∈ A | ∀a ∈ A, A (ba) = bA (a)}. We say that B ⊆ A is a coalgebra extension by C or a C-extension. The extension B ⊆ A is called a coalgebra-Galois extension (or a C-Galois extension) if the canonical left A-module, right C-comodule map canA : A ⊗B A → A ⊗ C,

a ⊗B a → aA (a ),

is bijective. The notion of a C-Galois extension in this generality and in this form was introduced in [46], extending the deﬁnition of a coalgebra principal bundle in [49]. In the case when C is a Hopf algebra and A is a C-comodule algebra (i.e. the coaction A is an algebra map), a coalgebra-Galois extension is known as a Hopf–Galois extension, a notion introduced in [127] (and rooted in an algebraic approach to Galois theory [73]). The discovery of examples of noncommutative spaces such as quantum spheres [155] made it clear that the notion of a Hopf–Galois extension is too rigid to capture all the geometrically interesting algebras, hence the introduction of a more general Galois-type extensions in [49]. 6.14. Canonical entwining for a coalgebra-Galois extension [46, Theorem 2.7]. Let A be a C-Galois extension of B. Then, there exists a unique entwining map C (ψ) by the structure maps μ and A . The ψ : C ⊗ A → A ⊗ C such that A ∈ MA A map ψ is called the canonical entwining map associated to a C-Galois extension B ⊆ A. Sketch of proof. The entwining map ψ is given by ψ : c ⊗ a → canA can−1 A (1A ⊗ c) a , where canA is the canonical map in 6.13.

6.15. Coalgebra-Galois extension as a Galois coring. By 6.14 and 3.21, if B ⊆ A is a C-Galois extension, then C = A ⊗ C is an A-coring with coproduct C = A ⊗ C , counit εC = A ⊗ εC , and A-multiplications a (a ⊗ c)a = a a canA can−1 A (1A ⊗ c) a . Since A is an entwined module, it is a right C-comodule; hence there exists a group-like element g; see 4.12. Since the C-coaction (induced

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by g) on A is the same as the C-coaction A , necessarily, g = A (1A ). Again, since A is an entwined module, one ﬁnds that C-coinvariants B can be described differently as B = {b ∈ A, | A (b) = bA (1A )}. Remember that the C-coaction on A is A (a) = ga. Hence b is an element of B if and only if gb = A (b) = bA (1A ) = bg. Thus B is the same subalgebra of A as g-coinvariants Ag in A. The canonical map canA in 6.13 coincides with the canonical map canA in 5.20. Therefore, if B ⊆ A is a C-Galois extension, then A is a Galois C-comodule [with respect to the group-like element A (1A )]. Conversely, if we are starting from an entwining structure (A, C)ψ such that C = A ⊗ C is a Galois A-coring, that is, A is a Galois comodule, then A is a C-Galois extension (of C-coinvariants). Thus the general theory of coalgebra-Galois extensions (understood as rudimentary noncommutative principal bundles) is a special case of the theory of (ﬁnite) Galois comodules. 6.16. Principal extensions. Although a coalgebra-Galois extension is a ﬁrst approximation to a noncommutative principal bundle, it does not have all desired geometric properties. One of missing properties is the existence of a connection or local triviality. The deﬁnition of a noncommutative principal bundle, which appears to ﬁll these gaps, was proposed in [47]. Since in differential geometry one is interested in algebras over a ﬁeld (typically the ﬁeld of complex numbers) in this item, we assume that k is a ﬁeld. A C-Galois extension B ⊆ A is called a principal extension if (a) the canonical entwining map ψ is bijective, (b) the multiplication map B ⊗ A → A has a left B-linear right C-colinear section (i.e. A is a C-equivariantly projective left B-module), (c) there exists a group-like element x ∈ C such that A (1A ) = 1A ⊗ x. From the algebraic point of view, (c) is a minor condition and is needed only for differential-geometric reasons. Since A is an entwined module, it is equivalent to the requirement that A (a) = ψ(x ⊗ a). If B ⊆ A is a principal extension, then A is a principal Galois comodule of the associated A-coring C = A ⊗ C (this is the original motivation for introducing principal Galois comodules over a coring). In geometrically interesting cases (e.g. when C is a compact quantum group), the coalgebra C is a coseparable coalgebra (i.e. it has a cointegral; cf. 5.3). Applying the structure theorem for principal comodules 5.26 to this particular setup, one ﬁnds that for an extension to be principal, it is sufﬁcient to assume that the canonical map canA is surjective. More precisely, there is: Theorem [40, Theorem 4.6], [166, Theorem 5.9]. Let k be a ﬁeld and (A, C)ψ an entwining structure such that the map ψ is bijective. Suppose that x ∈ C is a grouplike element and view A as a right C-comodule with the coaction A : A → A ⊗ C,

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a → ψ(x ⊗ a). If C is a coseparable coalgebra and the (lifted) canonical map A : A ⊗ A→A ⊗ C, can

a ⊗ a → aA (a ),

is surjective, then A is a principal C-extension of the coinvariants B. Principal extensions allow one to make a connection between the corepresentation theory of coalgebras C and the cyclic homology of B. This connection is provided by the Chern–Galois character. Its description goes beyond the scope of this chapter, and we refer the interested reader to the note [47] and to paper [26] for more detailed and general formulation.

6.3. Noncommutative algebraic geometry To the best of our knowledge, there are at least three approaches to noncommutative algebraic geometry in which corings play some role. The terminology is speciﬁc to each one of these approaches. 6.3.1. Approach through Grothendieck categories. This is an approach to noncommutative algebraic geometry advocated by Rosenberg [163, 164] and Van den Bergh [186]. Although corings do not seem to play as fundamental role as in two other approaches described in subsequent sections, the category of comodules appears to have a geometric meaning. Also, speciﬁc properties of the category of comodules have geometric interpretation. Here, we simply translate some of the properties of comodules into the language of noncommutative algebraic geometry. On the algebraic geometry side, we follow the terminology of [172, 173]. 6.17. Spaces as Grothendieck categories. A noncommutative space X is a Grothendieck category, denoted ModX. A standard classical (motivating) example of a space is the category of quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. Since a category of right modules over an algebra A is a Grothendieck category, it is a noncommutative space. A space X equivalent (as a category) to a module category MA is called a noncommutative afﬁne space. The algebra A is known as a coordinate ring of X. Making use of the Gabriel–Popescu theorem [156], one concludes that X is an afﬁne non-commutative space if and only if ModX has a ﬁnitely generated projective generator U. In this case A ModX(U, U ); see [174, p. 223]. 6.18. Maps between noncommutative spaces. A map X→Y between two noncommutative spaces is deﬁned as an adjoint pair of functors f = (f ∗ , f∗ ), f ∗ : ModY → ModX, f∗ : ModX → ModY . The functor f ∗ is referred to as the inverse image and f∗ is termed the direct image. The map f = (f ∗ , f∗ ) is said to be afﬁne if f∗ is faithful and has a right adjoint. 6.19. Comodules as a noncommutative space. Let C be an A-coring. In view of 4.30, if C is a ﬂat left A-module, then the category of right C-comodules MC

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is a Grothendieck category. Hence, in this case, MC is a noncommutative space. Furthermore, since the forgetful functor (−)A : MC → MA has the right adjoint −⊗A C : MA → MC , there is a morphism of noncommutative spaces ((−)A , −⊗A C) from MC to MA . If C is ﬁnitely generated and projective as a left A-module, then, in view of 4.38, MC is an afﬁne noncommutative space with coordinate ring B = (∗ C)op . In view of the ﬁnite Galois comodule structure theorem 5.22, if C is ﬂat as a left A-module and there exists a ﬁnite Galois comodule (M, M ) which is faithfully ﬂat as a left module over its coinvariant ring End C (M ), then MC is an afﬁne noncommutative space with the coordiante ring End C (M ). 6.20. Weakly closed subspaces. A subspace Y of a noncommutative space X is a full subcategory ModY of ModX, which is closed under direct sums, kernels, and isomorphisms. A subspace Y is said to be weakly closed if ModY is closed under subquotients and the inclusion functor ModY → ModX has a right adjoint; see [173, Deﬁnition 2.4]. For an A-coring C, MC = σ[C] if and only if C is a locally projective left A-module. Wisbauer’s σ-category is closed under direct sums, kernels, isomorphisms, and subquotients. Thus MC is the smallest weakly closed subspace of the afﬁne noncommutative space with the coordinate ring, B = (∗ C)op containing C if and only if C is a locally projective left A-module. 6.21. Integral spaces. A noncommutative space X is said to be integral if ModX = σ[E] for an indecomposable injective object E of ModX whose endomorphism ring is a division ring; see [172, Deﬁnition 3.1], [151, Deﬁnition 7.1]. Thus, if C is an A-coring, locally projective as a left A-module, indecomposable and injective as a C-comodule and such that the right dual algebra C ∗ is a division ring, then MC is an integral noncommutative space. Note that if C is a simple C-comodule, then it is indecomposable and C ∗ is a division ring (by the Schur lemma). On the other hand, any (locally projective as a left A-module) coring C over an injective ring A is injective in MC ; [53, 18.19]. In view of [151, Theorem 3.7], if C is ﬁnitely generated as a left A-module, then MC is an integral (afﬁne) space if and only if the left dual algebra ∗ C is a prime ring, that is C is a prime coring in the sense of [98]. 6.3.2. Approach through covers. This is an approach to noncommutative algebraic geometry in which corings play a central role. Initiated in [123] it is fully developed in the series of papers [124]. 6.22. Finite covers and quasi-coherent sheaves. An A-coring C that is faithfully ﬂat as a left A-module is called a ﬁnite cover. Covers are denoted by (A, C). Right C-comodules are called quasi-coherent sheaves. In the context of noncommutative geometry, MC is denoted by Qcoh(A, C). The category of covers Covers is deﬁned as a full subcategory of the opposite of the category of corings Crg consisting of all corings, which are faithfully ﬂat as left modules (over their base rings); compare 3.7. Thus a morphism of covers (A, C) → (B, D) is a pair of maps (α, γ), α : B → A, γ : D → C such that α is algebra map and A ⊗B γ ⊗B A is an A-coring map.

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6.23. Reﬁnements and space covers. Given an inclusion of algebras α : B → A and a B-coring D, consider the base algebra extension coring C = A ⊗B D ⊗B A; see 3.6. There is a coring morphism (γ, α) : D → C, where γ : d → 1A ⊗B d ⊗B 1A . If A and D are faithfully ﬂat as left B-modules, then C is faithfully ﬂat as a left A-module; hence it is a cover. In this case, the coring morphism (γ, α) induces a morphism of covers (A, C) → (B, D) (note the direction of the arrow). Any morphism of this type is called a reﬁnement morphism. The class of reﬁnement morphisms in Covers is denoted by Ref. A cover (A, C) is called a space cover if there is an A-coring epimorphism sC : A⊗A → C (where A⊗A is the Sweedler coring corresponding to 1A : k → A). In other words, a space cover is an A-coring C generated as a bimodule by a grouplike element and faithfully ﬂat as a left A-module. Space covers are denoted by (C, sC ). Morphisms of space covers are morphisms of covers compatible with structure maps sC , sD [i.e. sending the group-like element sC (1A ⊗ 1A ) to the grouplike element sD (1B ⊗ 1B )]. The resulting category of space covers is denoted by Coverssp . 6.24. Equivalent morphisms of covers and noncommutative spaces. Two mor ) : (C, s ) → (D, s ) of space covers are said to be equivalent phisms (α, γ), (α , γ C D if, for any element i xi ⊗ yi ∈ ker sD , α(xi )α (yi ) = α (xi )α(yi ) = 0. i

i

sp . A quotient category of Coverssp by this equivalence relation is denoted by Covers sp with A noncommutative space is deﬁned as an object of the localization of Covers respect to the class of reﬁnements Ref. 6.3.3. Approach through monoidal categories. This, the most recent proposal for noncommutative algebraic geometry, was made in [133] and is based on the premise that noncommutative schemes should be understood as monoidal categories; see 2.13. 6.25. Noncommutative schemes. Anoncommutative scheme is anAbelian monoidal category (Qcoh(X), X , OX ) ( X is multiplication, OX is a unit). A morphism of noncommutative schemes is an isoclass of an additive monoidal functor G : (Qcoh(X), X , OX ) → (Qcoh(Y ), Y , OY ), which has a left adjoint. If A is a k-algebra, then (A MA , ⊗A , A) is called a non-commutative afﬁne scheme. A noncommutative afﬁne scheme corresponding to A is denoted by Spec(A). 6.26. Morphisms of afﬁne schemes and representations. A B|A-coring C (cf. 3.27) is termed a representation of an algebra B over an algebra A. By the EilenbergWatts theorem (see [192, Theorem 6] or e.g. [174, Chapter IV, Proposition 10.1]), every additive functor between bimodule categories A MA → B MB , which has a left adjoint, can be represented as G = A HomA (C, −). Thus, by 3.27, there is a one-to-one correspondence between morphisms of afﬁne schemes and representations (or B|Acorings).

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6.27. Flat covers. In the set-up of 6.26, the left adjoint of G is the tensor functor F = − ⊗Bop ⊗B C. The A-coring structure on C induces an A-coring structure on D = F(B) = B ⊗Bop ⊗B C = C/[C, B], where [C, B] is the commutator submodule of C. For any B-bimodules M and N, the unit η and counit ψ of the adjunction (F, G) induce a morphism (natural in M and N) 2 FM,N

:

F(M ⊗B N ) F(G2F(M ),F(N ) )

F(ηM ⊗B ηN )

/ F(GF(M ) ⊗B GF(N ))

/ FG(F(M ) ⊗A F(N ))

ψF(M )⊗A F(N )

/ F(M )D F(N ) .

A morphism of afﬁne schemes G is said to be a ﬂat cover if the left adjoint F of G is 2 faithful and the maps FM,N are isomorphisms for all B-bimodules M, N. 6.28. Galois extensions. Let (D, ι) be a B|A-coring. The map ι : B → ∗ D∗ is called a noncommutative Galois ring extension if there exists an A-coring C and an A-coring morphism φ : D → C such that (a) the category of C-bicomodules is an Abelian monoidal category with multiplication C ; (b) (D ⊗A φ ⊗A D) ◦ ((D ⊗A D) − (D ⊗A D )) is a pure morphism of B-bimodules; (c) B is isomorphic to the algebra of coinvariants C HomC (D, C) (cf. 4.18); (d) B ⊗ D → DC D, b ⊗ d → d (1) ι(b)(d (2) ) ⊗A d (3) , is an isomorphism; (e) D is a faithfully ﬂat left Bop ⊗ B-module; (f) the map of A-corings φ¯ : D/[D, B] → C, ¯ deﬁned by φ = φ◦p, where p : D → D/[D, B] is the canonical epimorphism, is an isomorphism; (g) for all A-bimodules M, the map θM : A HomA (D, M) ⊗Bop ⊗B D → C ⊗A M ⊗A C, f ⊗ d → φ(d (1) ) ⊗A f(d (2) ) ⊗A φ(d (3) ), is an isomorphism (natural in M). From the point of view of Galois comodules studied earlier, the key (Galois) properties are (c) and (f ). More speciﬁcally, let C be an A-coring and (M, M ) be a right C-comodule that is ﬁnitely generated and projective as an A-module. Set B = End C (M), T = End A (M), and D = M ∗ ⊗ M [the (ﬁnite) comatrix coring]. Since M is a ﬁnitely generated and projective right A-module, ∗

D∗ = A HomA (M ∗ ⊗ M, A) End A (M ) = T,

so that there is an algebra inclusion ι : B → ∗ D∗ . Thus (D, ι) is a B|A-coring. Let φ : D = M ∗ ⊗ M → C, ξ ⊗ m → ξ(m(0) )m(1) ,

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be the (lifted) canonical map; see 5.26, 4.18. Then B = C HomC (D, C) (cf. 4.18); so condition (c) is automatically satisﬁed. Finally, D/[D, B] = M ∗ ⊗B M, p : M ∗ ⊗ M → M ∗ ⊗B M is the canonical surjection, hence φ¯ = canM . Thus, (f ) is the same as the ﬁnite Galois condition in 5.19. For a ﬁnite Galois comodule (M, M ), all other conditions follow, provided the functor M ∗ ⊗B − ⊗B M : B MB → A MA is faithful and exact, and the map T → T ⊗B T , t → 1T ⊗B t − t ⊗B 1T is pure in B MB . From the noncommutative algebraic geometry point of view, all the conditions (a)–(g) are needed, since as proved in [133, Theorem 3], there is a one-to-one correspondence between noncommutative Galois ring extensions and ﬂat covers in the category of noncommutative afﬁne schemes.

7. Extensions and dualizations 7.1. Weak and lax corings As explained in 3.21, if (A, C)ψ is an entwining structure, then A ⊗ C is an A-coring. On the other hand, if (A, C, ψ) is a weak entwining structure not all of A ⊗ C is a coring, but only a direct summand of it. Weak corings were introduced to answer the following question: what is a coring-like structure of A⊗C? On the other hand, recent interest in partial Galois theory led to the introduction of lax and partial entwining structures, and, consequently, lax corings. In this section, we brieﬂy describe these notions. 7.1. Weak corings. Let A be a k-algebra, and let C be an A-bimodule, which can be nonunital as both left and right A-module, that is, there can exist c ∈ C such that 1A c = c or c1A = c. C is called a weak A-coring [196] if there exist A-bimodule maps C : C → C ⊗A A ⊗A C and εC : C → A such that the following diagrams C

C C

C ⊗A A ⊗ A C

/ C ⊗A A ⊗ A C C⊗A A⊗A C

C ⊗A A⊗A C

/ C ⊗A A ⊗A C ⊗A A ⊗A C,

C N NNN ppp NNNC p p p NNN p p p N' p wp C ⊗A A ⊗NA C 1A ⊗A − −⊗A 1A C ⊗A A ⊗A C NNN pp NNN ppp p N p p CA⊗A εC εC ⊗A AC NNN N' wpppp C C

are commutative. Here 1A ⊗A − denotes the map c → 1A c and − ⊗A 1A is the map c → c1A . If C is a left (resp. right) unital A-bimodule, then we say that C is a left (resp. right) unital weak A-coring.

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For any weak A-coring C, ACA is a unital A-bimodule, and it is an A-coring. Similarly, if C is a left (resp. right) unital weak A-coring, then CA (resp. AC) is an A-coring. As for corings, the coproduct on C induces an associative multiplication on the dual modules C ∗ = HomA (C, A), ∗ C = A Hom(C, A), ∗ C ∗ = A HomA (C, A). ∗ C ∗ is a unital algebra with unit εC , and it coincides with the two-sided dual of the coring ACA. C ∗ and ∗ C are nonunital, but εC is a central idempotent in each one of them (hence they are ﬁrm rings). The dual rings of the coring ACA coincide with the unital subrings of dual rings of C generated (as modules) by εC . 7.2. Weak corings and weak entwining structures. Weak corings were introduced in [196] to understand fully the correspondence between weak entwining structures (see 3.22) and corings, in particular to obtain a statement similar to that in 3.21. This aim is achieved in the following Proposition. Let A be an algebra and (C, C , εC ) be a coalgebra. Set C = A ⊗ C and view it as a (unital) left A-module with the product a(a ⊗ c) = aa ⊗ c. If (A, C, ψ) is a weak entwining structure, then A ⊗ C is a (nonunital) right Amodule by (a ⊗c)a = a ψ(c⊗a), and C is a left unital weak A-coring with coproduct and counit C : A ⊗ C a⊗c

→

→

(A ⊗ C)1A ⊗ C, (A ⊗ C) ⊗A (A ⊗ C) (a ⊗ c(1) )1A ⊗ c(2) , (a ⊗ c(1) ) ⊗A (1A ⊗ c(2) ) →

εC : A ⊗ C a⊗c

→ (A ⊗ C)1A

→ (a ⊗ c)1A

→

→

A, (A ⊗ εC )((a ⊗ c)1A ).

Conversely, if C is a left unital weak A-coring with C and εC as deﬁned above, then C ⊗ A → A ⊗ C, c ⊗ a → (1A ⊗ c)a, is a weak entwining map for A and C. 7.3. Comments on weak comodules. Similarly as for corings, one deﬁnes comodules over a weak A-coring C or weak comodules. These are deﬁned as pairs (M, M ), where M is a nonunital right A-module, and M : M → M ⊗A AC is a right A-module map such that the following diagrams M M

M ⊗A AC

M

/

M ⊗A AC M⊗A C

/ M ⊗A AC ⊗A AC,

M ⊗A AC

M

/ M ⊗A AC M M MMM MMM M⊗A εC M −⊗A 1A MMM & M ⊗A A,

commute. Morphisms between comodules over weak corings are deﬁned in the same way as for comodules over corings. If (M, M ) is a right C-comodule, then (MA, M |MA) is a right comodule over the A-coring ACA. As a consequence of this, the category of weak comodules has similar properties to the category of comodules over a coring. For example, it always has cokernels and direct sums, and it has kernels provided that AC is a ﬂat left A-module. The tensor functor − ⊗A CA from the

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303

category of A-modules to C-comodules is the right adjoint of the functor − ⊗A A (the deﬁning adjunctions). For more detailed studies of weak corings and comodules, the reader is referred to [53, Chapter 6] and [196]. 7.4. Lax corings. Let A be a k-algebra, and let C be an A-bimodule, which is nonunital as a right A-module, that is, although it is required that, for all c, 1A c = c, it is not required that c1A = c. Write C for C1A = CA. C is a unital A-bimodule, and it is a direct summand of C as an A-bimodule. Denote the inclusion C ⊆ C by ι. Since C is a unital left A-module, any A-bimodule map C : C → C ⊗A C restricts to the map C : C → C ⊗A C. C is called a left unital lax A-coring [63] if there exist a coassociative A-bimodule map C : C → C ⊗A C and an A-bimodule map εC : C → A such that C is an A-coring with coproduct C and counit εC ◦ ι. A left unital weak A-coring is an example of a left unital lax A-coring. 7.5. Lax and partial entwining structures and corings. Let A be an algebra and C be a coalgebra. Consider a map ψ: C ⊗ A → A ⊗ C, and write, for all c ∈ C, a ∈ A, ψ(c ⊗ a) = ψ aψ ⊗ cψ = a ⊗ c . Assume that for all a, b ∈ A, and c ∈ C, (ab)ψ ⊗ cψ = aψ b ⊗ cψ ,

ψ

aψ 1A ⊗ c

ψ

ψ,

(1)

⊗c

ψ

(2)

ψ,

=

aψ ⊗ c(1) ⊗ c(2) ψ .

ψ,

The ﬁrst of these conditions means that A ⊗ C is a nonunital right A-module with multiplication (a ⊗ c)a = a ψ(c ⊗ a). Write C = A ⊗ C and deﬁne C : A ⊗ C → A ⊗ C ⊗A A ⊗ C (A ⊗ C)1A ⊗ C, a ⊗ c → aψ(c(1) ⊗ 1A ) ⊗ c(2) . If the map ψ satisﬁes the following additional conditions, for all a ∈ A, c ∈ C, εC (cψ )1Aψ a = εC (cψ )aψ , 1Aψ ⊗cψ = εC (c(1) )1Aψ ⊗c(2) ψ , ψ

ψ

ψ

ψ,

then the triple (A, C, ψ) is called a lax entwining structure. The existence of a lax entwining structure is equivalent to the statement that C is a left unital lax A-coring with coproduct C and counit εC : A ⊗ C → A, a ⊗ c → ψ a1Aψ εC (cψ ) (and the canonical left A-multiplication). If the map ψ satisﬁes the following additional condition, (εC ⊗ A) = (A ⊗ εC ) ◦ ψ, then the triple (A, C, ψ) is called a partial entwining structure. The existence of a partial entwining structure is equivalent to the statement that C is a left unital lax A-coring with coproduct C and counit A ⊗ εC .

304

T. Brzezi´nski

Given a lax entwining structure (A, C, ψ), a lax entwined module is a right A-module M together with a k-linear map M : M → M ⊗ C, written on elements M as (m) = m[0] ⊗ m[1] , such that, for all m ∈ M, a ∈ A, (a) (M ⊗ εC ) ◦ M = M, (b) (M ⊗ C) ◦ M (m) = ψ, m[0] 1Aψ ⊗ m[1] (1) ⊗ m[1] (2) ψ , (c) M (ma) = m[0] ψ(m[1] ⊗ a). The category with objects lax entwined modules and morphisms A-linear, C-colinear C (ψ)lax . This category is isomorphic to the category MC , where maps is denoted by MA C = A ⊗ C := (A ⊗ C)1A is the A-coring associated to a left unital lax A-coring C = A ⊗ C; see 7.4. 7.6. Partial entwining structures and partial (co)actions. Let H be a bialgebra. (1) An algebra A is said to be a partial right H-comodule algebra if there exists a k-linear map A : A → A ⊗ H, written on elements as A (a) = a[0] ⊗ a[1] , such that, for alla, b ∈ A, (a) A (ab) = a[0] b[0] ⊗ a[1] b[1] , [0] (b) a εH (a[1] ) = a, A [0] (c) (a ) ⊗ a[1] = a[0] 1A [0] ⊗ a[1] (1) 1A [1] ⊗ a[1] (2) . (2) A coalgebra C is said to be a partial right H-module coalgebra if there exists a nonunital right action of H on C, C : C ⊗ H → C, c ⊗ h → ch, such that, for all c ∈ C and h ∈ H, (a) εC (ch) = εC (c)εH (h), (b) (ch)(1) 1H ⊗ (ch)(2) = c(1) h(1) ⊗ c(2) h(2) . If A is a partial right H-comodule algebra and C is a partial right H-module coalgebra, then there is a partial entwining structure (A, C, ψ), where ψ : C ⊗ A → A ⊗ C, c ⊗ a → a[0] ⊗ ca[1] . Partial entwining structures of this (or its dual) kind arise in the context of partial group actions; see [63, Section 5], [57, 85]. 7.7. Comments on the structure and usage of lax corings. Since lax corings are deﬁned through corings, that is, C is a left unital lax coring if and only if C is an A-coring, they share many properties with the latter. For example, their dual modules are (nonunital) rings. Using this correspondence, one can study Frobenius or Galois properties of lax corings. In the case of lax corings associated to partial and lax entwining structures, where the category of lax entwined modules is isomorphic to the category of comodules of the associated A-coring A ⊗ C, such properties are a consequence of the (nonlax) coring theory. For further details, the reader is referred to [63]. 7.2. C-rings A notion of a C-ring or a C-algebra arises as a dualization of that of coring; corings are coalgebras in the category of bimodules, C-rings are algebras in the category of

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305

bicomodules. To avoid complications arising from the nonassociativity of cotensor product (for coalgebras over rings), in this section, we assume that k is a ﬁeld. In this case, given a coalgebra C, for all C-bicomodules (M, M , M), (N, N , N), the cotensor product MC N is a C-bicomodule with coactions MC N and MC N. Furthermore, for any bicomodule morphisms f , g, also f C g is a bicomodule morphism (i.e. −C − is a functor C MC × C MC → C MC ), and the category of C-bicomodules is a monoidal category, (C MC , C , C); see 2.13. 7.8. C-rings [37, Section 6]. Let C be a k-coalgebra and let A be a C-bicomodule with coactions A : A → C ⊗ A and A : A → A ⊗ C. A is called a C-ring if there are two bicomodule maps μA : AC A → A and ηA : C → A rendering the following diagrams commutative AC AC A

μA C A

/ AC A μA

AC μA

AC A

A

A

A

Ao

μA

/ A,

μA

/ AC C

AC ηA

AC A ,

A

A

A

Ao

μA

/ CC A

ηA C A

AC A .

Equivalently, a C-bicomodule A is a C-ring if and only if the cotensor functor −C A : MC → MC (or, equivalently, AC − : C M → C M) is a monad. Still equivalently, a C-ring is a monoid (algebra) in the monoidal category (C MC , C , C). C-rings in this form were introduced in [37] to retain properties of entwining structures, which are not captured by corings, and in [157] as a natural algebraic structure underlying semi-inﬁnite cohomology [190]. In [157], C-rings are termed C-algebras. However, since in parallel to the k-algebra case, the term “C-algebra” might suggest that A is C-(co)central, the term “C-ring” is our preferred name. 7.9. Modules over C-rings. A right module over a C-ring A is a triple (M, M , M ), where (M, M ) is a right C-comodule M with coaction M : M → M ⊗ C, and M : MC A → M is a right C-comodule map such that MC AC A

M C A

M

MC μA

MC A

/ MC A

M

/ M,

M

M

M

Mo

M

/ MC C

MC ηA

MC A .

The map M is called a right A-action. A map of A-modules is a right C-colinear map, which respects the actions, that is, a map of A-modules (M, M , M ) → (N, N , N )

306

T. Brzezi´nski

is a C-comodule map f : M → N such that N ◦ (f C A) = f ◦ M . The category of right A-modules is denoted by MA . The category of right A-modules coincides with the category of algebras of the monad −C A : MC → MC , that is, MA = (MC )−C A . Consequently the functor −C A : MC → MA ,

(M, M ) → (MC A, MC A , MC μA ),

is a left adjoint of the forgetful functor (−)C : MA → MC . Left A-modules are deﬁned symmetrically as triples (N, N, N ), where (N, N) is a left C-comodule and N : AC N → N is an associative, unital left C-comodule map. 7.10. Separable and split C-rings [37, Theorem 6.3]. The deﬁning adjunction −C A, (−)C retains large amount of information about the structure of C-rings. For example, the functor −C A : MC → MA is separable if and only if there exists a map e : A → k such that (e ⊗ C) ◦ ρA = (C ⊗ e) ◦ A and e ◦ ηA = εC . A C-ring with such a map is called a split C-ring. The forgetful functor (−)C : MA → MC is separable if and only if there exists γ ∈ C HomC (C, AC A) such that (μA C A) ◦ (AC γ) ◦ A = (AC μA ) ◦ (γC A) ◦ A and μA ◦ γ = ηA . In this case, A is called a separable C-ring. A right A-module (P, P , P ) is said to be (A, C)-projective if the action P has an A-module section. Every module of a separable C-ring A is (A, C)-projective. 7.11. Characters. A character in A is a k-linear map κ : A → k such that κ ◦ μA = κC κ,

κ ◦ ηA = εC .

The regular right C-comodule (C, C ) is a right A-module if and only if there is a character in A. As explained in [43, Section 8], to every C-ring A with a character, one can associate a comodule with an extended coderivation (cf. [82]) and then interpret right A-modules as comodules with ﬂat connections. This is parallel to the description of comodules over a coring with a group-like element; see 4.23. 7.12. Trivial and dual-Sweedler C-rings. A coalgebra C is itself a C-ring with both the unit and multiplication provided by the identity map C (combined with the isomorphism CC C C in case of the multiplication). Right C-modules are the same as right C-comodules, so MC ≡ MC . Let π : C → B be a morphism of coalgebras, then C is a B-bicomodule via (C ⊗ π) ◦ and (π ⊗ C) ◦ . Deﬁne A = CB C. Then, A is a C-ring with product μA : AC A ∼ = CB CB C → A, μA = CB εC B C, and unit νA = C . This is a dualization of the Sweedler coring. The objects in the category of right A-modules are triples (M, M , f ), where (M, M ) is a right C-comodule and f : MB C → M

Comodules and Corings

307

is a right C-comodule map such that f ◦ (f B C) = f ◦ (MB εC B C),

f ◦ M = M.

By analogy with Sweedler’s coring and descent data, it appears quite natural to call such triples (M, M , f ) ascent data. 7.13. C-rings and entwining structures. The main motivation for introducing C-rings in [37] was that their deﬁning adjunction involves the forgetful functor (−)C : MA → MC . If, therefore, there is a C-ring associated to an entwining C (ψ) (the category of entwined modules), then, structure (A, C)ψ and MA ≡ MA by studying general properties of C-rings, one derives these properties of entwined modules, which arise from the properties of (−)C . This is, indeed, the case. Proposition. For an entwining structure (A, C)ψ , view A = C ⊗ A as a C-bicomodule with the left coaction A = C ⊗ C and the right coaction A = (C ⊗ ψ)◦(C ⊗A). Then A is a C-ring with the product μA : AC A ∼ = C ⊗A⊗A → A, μA = C ⊗ μA , and the unit ηA = C ⊗ 1A . Conversely, if C ⊗A is a C-ring with the product and the unit above and the natural left C-comodule structure C ⊗ A, then (A, C)ψ is an entwining structure, where ψ = (εC ⊗ A ⊗ C) ◦ C⊗A . C (ψ). Under this bijective correspondence, MA ≡ MA In a similar way, one associates a C-ring to a weak entwining structure (A, C, ψ). In this case, A = Im pR , where pR : C ⊗ A → C ⊗ A,

pR = (C ⊗ A ⊗ εC ) ◦ (C ⊗ ψ) ◦ (C ⊗ A),

and the unit of A is c → pR (1A ⊗ c); see [51, Section 4]. 7.14. Matrix ring contexts [51]. A matrix ring context, (C, D, N, M, σ, τ), consists of a pair of coalgebras C and D, a (C, D)-bicomodule (N, N , N), a (D, C)bicomodule (M, M , M), and a pair of bicomodule maps σ : C → ND M,

τ : MC N → D,

such that the diagrams ND MC N o ND τ

σC N

CC N

O

N

ND D o

N

N,

MC ND M

O

τD M

M

MC σ

MC C o

/ DD M O

M

M

commute. The map σ is called a unit and τ is called a counit of a matrix context. Since a counit τ of a matrix ring context is a D-bicomodule map, it is fully deterτ is called a reduced counit of a mined by its reduced form τ = εD ◦ τ. The map matrix context. The D-bicolinearity of τ is equivalent to the following property of τ, τ ⊗ D) ◦ (M ⊗ N ). (D ⊗ τ ) ◦ (M ⊗ N ) = (

308

T. Brzezi´nski

In terms of the reduced counit, the above commutative diagrams read (N ⊗ τ ) ◦ (σC N ) ◦ N = N,

( τ ⊗ M) ◦ (MC σ) ◦ M = M.

The notion of a matrix context is closely related to that of pre-equivalence data or a Morita-Takeuchi context introduced in [180, Deﬁnition 2.3]. In particular, in view of [180, Theorem 2.5], if one of the maps in a Morita-Takeuchi context is injective, then there is a corresponding matrix ring context. Furthermore, all equivalence data give rise to a matrix ring context. Matrix ring contexts are in one-to-one correspondence with adjoint pairs of functors −C N : MC → MD , −D M : MD → MC (or adjunctions in a bicategory: coalgebras, bicomodules, bicolinear maps; for bicategories see [19]). In particular, if (C, D, N, M, σ, τ) is a matrix ring context, then M is a quasi-ﬁnite injector, that is, the cotensor functor −C N : MC → Mk is a left adoint of the tensor functor G = − ⊗ M : Mk → MC ([180, Proposition 1.3], [53, Section 12.8]). Conversely, (M, M ) can be a part of a matrix ring context only if it is a quasi-ﬁnite injector. 7.15. Matrix C-rings. Let (C, D, N, M, σ, τ) be a matrix ring context. Then A := ND M is a C-ring with the product μA = ND τ D M and unit ηA = σ, where τ is the reduced counit. Furthermore, M is a right A-module with the action τ D M τ. The C-ring A is called a matrix and N is a left A-module with the action ND C-ring. The monad −C A : MC → MC is simply the monad corresponding to the adjoint pair of functors −C N : MC → MD , −D M : MD → MC ; see 7.14. 7.16. Tensor product of modules and the coendomorphism C-ring. Let A be a C-ring, (M, M , M ) a right A-module, and (N, N, N ) a left A-module. The tensor product of (M, M , M ) with (N, N, N ) is deﬁned as the coequalizer of k-linear maps MC AC N

M C N MC N

/

/ MC N

πM,A,N

/

M ⊗A N,

that is, M ⊗A N is the cokernel of M C N − MC N . As argued in [157], semiinﬁnite cohomology of [11, 190] can be understood in terms of derived functors of such a tensor product for a speciﬁc C-ring. Let (M, M ) be a quasi-ﬁnite injector, and let (N, N) be a left C-comodule such that −C N is the left adjoint of the tensor functor − ⊗ M : Mk → MC . Then E = MC N is a coalgebra, known as a C-coendomorphism coalgebra of M. (E arises as a comonad − ⊗ MC N : Mk → Mk corresponding to the adjoint pair of functors −C N, − ⊗ M.) Furthermore, there exists a unique coalgebra structure on M ⊗A N, whereby πM,A,N is a coalgebra map. The coalgebra M ⊗A N is called an A-coendomorphism coalgebra of M and is denoted by EA (M). To any right A-module (M, M , M ), which is a quasi-ﬁnite injector as a right C-comodule, one can associate the matrix ring context (C, EA (M ), N, M, σ, τ) with ηC

σ : C −→ NE M → NEA (M) M,

ψE

τ : MC N −→ E − → EA (M),

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where η is the unit and ψ is the counit of adjunction −C N : MC → ME , −E M : ME → MC . We refer to this context as an A-coendomorphism context associated to M. The corresponding matrix C-ring is called an A-coendomorphism ring of M. 7.17. Galois theory. In parallel to (ﬁnite) Galois comodules of a coring, one can study Galois comodules of a C-ring A. Let (M, M , M ) be a right A-module such that (M, M ) is a quasi-ﬁnite injector. Set (N, N) to be a left C-comodule such that −C N is the left adjoint of the tensor functor − ⊗ M : Mk → MC . Let S = EA (M ) and let σ be the unit of the A-coendomorphism context associated to M. Then M is called a Galois A-module if the map β : A → NS M,

β := (N ⊗ M ) ◦ (σC A) ◦ A,

is an isomorphism. As shown in [51, Theorem 3.11], M is a Galois A-module if and only if β is a split monomorphism of left A-modules. 7.18. Bicoalgebroids. Bialgebroids are deﬁned as corings with a compatible algebra structure. In a dual way, one can deﬁne bicoalgebroids as C-rings with compatible coproduct and counit. Let (C, C , εC ) and (H, H , εH ) be coalgebras, and assume that there is a coalgebra map α : H → C and ananticoalgebra map β : H → C such that for all α(h(2) ) ⊗β(h(1) ). This allows one to view H as h ∈ H, α(h(1) ) ⊗ β(h(2) ) = H (h) = α(h(1) ) ⊗ h(2) and the right coaction a C-bicomodule via left coaction H (h) = h(2) ⊗ β(h(1) ). Following [50], H is called a bicoalgebroid if (a) (H, μH , ηH ) is a C-ring; (b) for all i gi ⊗ hi ∈ HC H, μH (gi ⊗ hi (1) ) ⊗ α(hi (2) ) = μH (gi (1) ⊗ hi ) ⊗ β(gi (2) ), i

H (μH (

i

gi ⊗ hi )) =

i

μH (gi (1) ⊗ hi (1) ) ⊗ μH (gi (2) ⊗ hi (2) ),

i

and εH C εH = εH ◦ μH ; (c) for all c ∈ C, εH (ηH (c)) = εC (c), and H (ηH (c)) = ηH (c)(1) ⊗ηH (α(ηH (c)(2) )) = ηH (c)(1) ⊗ηH (β(ηH (c)(2) )). Similarly to bialgebroids, the category of left comodules of a C-bicoalgebroid H is monoidal and the forgetful functor to the category of C-bicomodules is strict monoidal [13]. 7.3. Other topics As time, space, and the author’s expertise are all limited, there are several topics in coring and comodule theory, which are not covered in this chapter. In this ﬁnal section, we would like to list some of these topics and supply the reader with references.

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7.19. Contramodules. While we concentrated on comodules, there is another, much less-studied but seemingly equally natural, category of representations of corings. Let C be an A-bimodule. Since the tensor functor −⊗A C : MA → MA has a right adjoint, the hom-functor HomA (C, −), the characterization of corings as comonads −⊗A C in 3.5 can be extended to the characterization of corings as monads HomA (C, −). This has been observed already in [90]. As explained in 4.7, right comodules over C are the same as coalgebras of the comonad − ⊗A C. Algebras of the monad HomA (C, −) are known as contramodules over C [89]. Various properties of contramodules are discussed in [29, 157]. 7.20. Module theoretic and topological aspects. Several module theoretic and topological aspects of corings are described in [53]. Zariski topology for corings and comodules and top bicomodules are studied in [6]. Rational modules, pairings, and related topologies are described in [2]. Further rationality properties and comodules over semiperfect corings are studied in [62, 69]. (Co)prime and (co)semiprime corings and comodules are discussed in [5, 98, 193, 194]. Symmetric corings are considered in [16]. 7.21. Applications to representation theory. Some of the earliest applications of corings were in representation theory, in particular to matrix problems (cf. [161]). In representation theory, corings are known as BOCSs, and among others, they were used in settling the tame-wild conjecture [86]. Possibly the earliest reference to BOCSs is [162], and their role in the tame-wild conjecture is explained in [78, 87]. 7.22. Index theory for C ∗ -algebras. Corings appeared in [191] as an algebraic structure describing (ﬁnite) index theory for operator algebras [114]. Motivated by this, coring structures on Hilbert C∗ -modules are studied in [147, 148]. The theory of depth-2 extensions initiated as a special case of the Jones index theory in [118] makes extensive use of corings (bialgebroids); see also 3.19. 7.23. Categorical aspects. Throughout the chapter we have not discussed in great detail the categorical aspects of corings.An algebraist oriented overview of categorical aspects of corings can be found in [200]. One of the main modern examples of corings is a coring associated to an entwining structure. An entwining structure itself is an example of a distributive law [18]. For recent developments in the categorical approach to distributive laws, we refer to [171, Section 6], [170, 200]. Entwining structures over noncommutative rings and associated corings are studied in [24]. For a description of functors (and equivalences) between categories of comodules, see, for example, [4, 95, 101, 187]. Since, from the categorical point of view, corings are comonads (in a speciﬁc category), they ﬁt well into the formal theory of (co)monads [128, 175]. The bicategories arising from this formal theory are discussed in [44]. Monoidal aspects of the category of corings are discussed in [91]. 7.24. The Picard and Brauer groups. The Picard and Brauer groups for corings and Azumaya corings are studied in [60, 202]. Corings are used in studies of Picard and Brauer groups for rings in [61, 94, 134, 183].

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7.25. Cohomology. Cohomology of corings and comodules is discussed in [97, ˇ 106]. Cyclic cohomology of corings is studied in [31, 160]. Cech cohomology for corings is proposed in [169]. This construction is motivated by the interpretation of complete coverings of algebras [70] in terms of Galois corings [54]. Non-Abelian (descent) cohomology that extends non-Abelian cohomology of Hopf algebras [146] is proposed in [42] and is given a categorical treatment in [139]. Some cohomological aspects of corings are also described in [53, Section 30]. 7.26. Generalizations of corings. Corings with local structure maps are introduced in [188] and group corings in [64]. 7.27. Universal algebra and computer science. Coalgebras in universal algebra, and therefore also corings, are used in theoretical computer science. For example, entwining structures appear in [185] in the context of operational semantics. The interested reader can ﬁnd some comments and references in [200]. Acknowledgments I would like to thank Gabriella Böhm, Stefaan Caenepeel, José Gómez-Torrecillas, Joost Vercruysse, and Robert Wisbauer for many valuable comments and suggestions. References [1] Abrams, L., Two-dimensional topological quantum ﬁeld theories and Frobenius algebras, J. Knot Theory Ramiﬁ. 5 (1996), 569–587. [2] Abuhlail, J., Rational modules for corings, Comm. Algebra 31 (2003), 5793–5840. [3] Abuhlail, J., Morita contexts for corings and equivalences, in: S. Caenepeel, F. Van Oystaeyen eds., Hopf Algebras in Noncommutative Geometry and Physics. Lecture Notes in Pure and Appl. Math., 239, Marcel Dekker, New York, 2005, pp. 1–19. [4] Abuhlail, J., A note on coinduction functors between categories of comodules for corings, Annali dell’Universitá di Ferrara, Sezione VII.-Sci. Mat. 51 (2005), 151–172. [5] Abuhlail, J., Fully coprime comodules and fully coprime corings, Appl. Categ. Str. 14 (2006), 379–409. [6] Abuhlail, J., A Zariski topology for bicomodules and corings, Appl. Categ. Str. 16 (2008), 13–28. [7] Abuhlail, J., Gómez-Torrecillas, J., Lobillo, F.J., Duality and rational modules in Hopf algebras over commutative rings, J. Algebra 240 (2001), 165–184. [8] Aguiar, M., Internal Categories and Quantum Groups, PhD Thesis, Cornell University, 1997. [9] Anderson, F., Fuller, K., Rings and Categories of Modules, Springer, Berlin, 1974. [10] Ardizzoni, A., Böhm, G., Menini, C., A Schneider type theorem for Hopf algebroids, J. Algebra 318 (2007), 225–269. [11] Arkhipov, S.M., Semi-inﬁnite cohomology of associative algebras and bar duality, Internat. Math. Res. Not. 17 (1997), 833–863. [12] Baez, J. This week’s ﬁnds in Mathematical Physics (week 174), http://math.ucr.edu/home/baez/ week174.html (2001), last accessed on 25 February, 2009. [13] Bálint, I., Scalar extension of bicoalgebroids, Appl. Categ. Str. 16 (2008), 29–55. [14] Bass, H., Algebraic K-Theory, W.A. Benjamin, Inc., New York, 1968. [15] Baum, P.F., Hajac, P.M., Matthes, R., Szyma´nski, W., Noncommutative geometry approach to principal and associated bundles, arXiv:math/0701033 (2007).

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Witt vectors. Part 11 Michiel Hazewinkel Burg. s’Jacob laan 18, 1401BR Bussum, The Netherlands E-mail: [email protected]

Contents 1. Introduction and delimitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Historical motivation for the p-adic Witt polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Historical motivation for the (generalized) Witt polynomials . . . . . . . . . . . . . . . . . . . . . 1.11. Representability of the Witt vector functor. Symmetric functions . . . . . . . . . . . . . . . . . . 1.14. The plethora of structures on Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15. Initial sources of information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The p-adic Witt vectors: More historical motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Teichmüller representatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Construction of the functor of the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The p-adic Witt polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. ‘Miracle’ of the p-adic Witt polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Congruence formulas for the p-adic Witt polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Witt vector addition and multiplication polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14. Existence theorem for the p-adic Witt vectors functor . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16. Ghost component equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19. Ideals and topology for Witt vector rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.21. Teichmüller representatives in the Witt vector case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22. Multiplication with a Teichmüller representative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.25. Verschiebung on the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27. Frobenius on the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.37. Taking p-th powers on the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.40. Adding “disjoint” Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.42. Product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.44. fp Vp = [p] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Part 2 (and last) is planned for volume 7 of the Handbook of Algebra.

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00207-6 319

325 325 326 327 327 328 329 329 331 331 331 332 332 333 333 334 334 334 335 335 335 336 337 337 338

320

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6. The ring of p-adic Witt vectors over a perfect ring of characteristic p . . . . . . . . . . . . . . . . . . . . 6.1. The p-adic Witt vectors over a ring of characteristic p . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. The Valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10. p-adic valuation on the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13. The p-adic Witt vectors over a perfect ring of characteristic p . . . . . . . . . . . . . . . . . . . . 6.14. Universality property of the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19. Uniqueness of the p-adic Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20. The p-adic Witt vectors over a ﬁeld of characteristic p. Valuation on Wp∞ (k) . . . . . . . . 6.23. Cohen rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.24. p-basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.25. The Cohen functor of Colette Schoeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.26. Cohen ring of a ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Cyclic Galois extension of degree pn over a ﬁeld of characteristic p > 0 . . . . . . . . . . . . . . . . . 7.1. Construction of some Abelian extensions of a ﬁeld of characteristic p > 0 . . . . . . . . . . . 7.6. The ‘Artin-Schreier’ operator p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.9. The main theorems for Abelian extension of degree pn of ﬁelds of characteristic p . . . . 7.10. Kummer theory theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11. Cyclic extensions of degree pn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.12. On the proofs of 7.10 and 7.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Cyclic central simple algebras of degree pn over a ﬁeld of characteristic p > 0 . . . . . . . . . . . . 8.1. Central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Crossed product central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7. Cyclic central simple algebra associated to a ﬁnite length Witt vector . . . . . . . . . . . . . . . 8.10. Brauer group and Witt vectors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. The functor of the big Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. The functor of one power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7. Ghost component deﬁning formula for power series . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10. The symmetric funtion point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.15. Multiplication on the (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.20. The functor of the big Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.25. The Witt polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.28. Witt coordinates versus power series coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.29. Very partial symmetric function formularium (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.30. Partitions and monomial symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.32. Monomial symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.35. Complete symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.37. Wronsky relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.38. Symm ⊂ Z[ξ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.40. Hall inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.42. Inner product theorem. Dual orthonormal bases expansion . . . . . . . . . . . . . . . . . . . . . . . 9.45. The Schur symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.51. Jacobi-Trudy formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.52. Schur identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.53. Forgotten symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.54. Power sum symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.57. Newton relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.63. Witt coordinate symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.69. Power sum plethysm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.71. Duality formula for the Witt coordinate basis functions . . . . . . . . . . . . . . . . . . . . . . . . . 9.73. Big Witt verctor polynomial miracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.75. Big Witt vector addition and multiplication formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.89. Functional equation integrality lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

338 338 339 339 340 340 341 342 342 343 343 344 344 344 345 346 346 346 346 346 346 347 347 348 348 348 349 350 350 350 351 352 353 353 353 354 354 354 354 355 355 355 355 356 357 357 357 358 359 359 359 359 361

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11.

12.

13.

9.93. Ghost component integrality lemma. Ghost Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . 9.95. Integrality condition over the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.97. Witt vectors over the integers and ﬁxed points and ﬁxed point indices of iterated mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hopf algebra Symm as the representing algebra for the big Witt vectors . . . . . . . . . . . . . . . 10.7. Skew Schur functions. (Very partial symmetric functions formularium (2)) . . . . . . . . . . 10.12. Grading Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.14. Antipode on connected graded bialgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.15. Primitives of Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.18. Liulevicius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.23. Product comultiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.25. Various descriptions of the second comultiplication on Symm . . . . . . . . . . . . . . . . . . . . 10.26. Product comultiplication vs inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QSymm, the Hopf algebras of quasisymmetric functions, and NSymm, the Hopf algebra of non commutative symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. The Hopf algebra of non commutative symmetric functions . . . . . . . . . . . . . . . . . . . . . . 11.8. Divided power sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.13. Universal curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.14. NSymm represents curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.15. Second universality property of the Witt vectors: Symm represents curves in the commutative case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.17. The Hopf algebra QSymm of quasi-symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . 11.18. Compositions and partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.19. Monomial quasi-symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.26. Overlapping shufﬂe product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.29. Hopf algebra structure on QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.33. Duality between NSymm and QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.36. Primitives of QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.37. Autoduality of Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.39. Second comultiplication on QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.42. Distributivity of μP over μS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.45. Co-unit for the second comultiplication on QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.47. Second multiplication on NSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.48. Unit for the second multiplication on NSymm and Symm . . . . . . . . . . . . . . . . . . . . . . . Free, cofree, and duality properties of Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. Free algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4. Tensor algebra and tensor coalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8. The graded case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.11. Cofree coalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.15. Inheritance of structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.22. Free(CoFree(Z)) and Freecomm (CoFree(Z)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frobenius and Verschiebung and other endomorphisms of and the Witt vectors . . . . . . . . . . . 13.1. Verschiebung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5. Verschiebung is additive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6. Frobenius operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.13. Congruence properties of the Frobenius deﬁning polynomials . . . . . . . . . . . . . . . . . . . . 13.18. Ghost component characterization of the Frobenius operations . . . . . . . . . . . . . . . . . . . 13.19. Frobenius-like property of the Frobenius operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.22. The Frobenius operations are the unique functorial ring homomorphisms . . . . . . . . . . . . 13.33. Frobenius endomorphisms of QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.37. Frobenius on the monomial symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.39. No Frobenius endomorphisms of NSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.40. Frobenius like property of the f p on QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

321 361 362 362 363 364 364 365 365 365 366 367 368 369 369 370 370 370 371 371 371 371 372 373 373 374 374 374 375 376 377 377 377 378 378 378 379 380 381 382 382 382 383 383 383 384 385 386 387 387 387

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13.42. Multiplication by n operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.45. Homothety operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.48. Relations between operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.73. Frobenius and Verschiebung nomenclature convention . . . . . . . . . . . . . . . . . . . . . . . . . . 13.74. Ring of Hopf endomorphisms of Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. Supernatural and other quotients of the big Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5. Nests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7. Supernatural numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8. Supernatural quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9. Operations on nested Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11. Möbius function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.21. p-typiﬁcation for the big Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.24. p-typiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.25. Decomposition of the big Witt vectors over Z(p) -algebras . . . . . . . . . . . . . . . . . . . . . . . 14.27. Iterated nested Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. Cartier algebra and Dieudonné algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1. (h-h)-matrix of an endomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5. DE-matrix of an operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.12. Proof of the DE-principle (splitting principle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.16. Cartier algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.20. Witt vectors as endomorphisms of Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.23. W(R) ⊂ End(R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.24. Further description of the Cartier algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.25. The ring of additive endo operations of Wp∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.27. Endomorphisms of the inﬁnite dimensional additive group . . . . . . . . . . . . . . . . . . . . . . 15.32. End Wp∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.42. Dieudonné algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.45. End(Wpn ) over a perfect ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.46. Hirzebruch polynomials. (Very partial symmytric functions formularium (3)) . . . . . . . . 15.54. Open problem on endomorphisms of Symm as Witt vectors over the polynomials . . . . . 16. More operations on the and W functors: λ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1. Third universality property of the Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2. Fourth universality property of the Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3. Fifth universality property of the Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4. Deﬁnition of λ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.14. λ-rings vs σ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.19. σ-ring structure on (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.20. Adams operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.22. First Adams = Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.34. Existence lemma (for exterior powers and other structures) . . . . . . . . . . . . . . . . . . . . . . 16.40. Determinantal formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.42. Clarence Wilkerson theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.43. (A) is a λ-ring. Deﬁnition of the Artin-Hasse exponential . . . . . . . . . . . . . . . . . . . . . . 16.45. Deﬁning equations of the Artin-Hasse exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.46. λ-ring structure on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.47. The essence of a λ-ring structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.48. -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.50. Frobenius morphism like property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.51. -ring with Frobenius morphism like property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.52. Adams operation criterium for the λ-ring property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.53. Noncommutative -rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.54. Monads and comonads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

388 388 388 391 392 393 393 394 394 394 395 396 397 397 398 398 398 399 400 401 402 403 404 404 404 405 407 407 407 409 409 409 410 410 410 411 412 412 413 413 414 414 415 415 416 416 417 417 418 418 418 418

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16.59. Comonad structure on the Witt vectors. Characterization of λ-rings as the coalgebras for this comonad. (Fifth universality property of the Witt vectors) . . . . . . . . . . . . . . . . . 16.62. Cofreeness of the Witt vector rings (Third universality property of the Witt vectors) . . . 16.65. λ-ring structure on Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.69. λ-ring structure on QSymm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.70. Lyndon words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.71. Free polynomial generators for QSymm over the integers . . . . . . . . . . . . . . . . . . . . . . . 16.72. Discussion. The many operations on Symm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.73. Second Adams = Frobenius theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.74. Universal λ-ring on one generator (Fourth universality property of the Witt vectors) . . . 16.75. Relations between the third, fourth, and ﬁfth universality properties . . . . . . . . . . . . . . . . 16.76. Plethysm (Very partial symmetric function formularium (4)) . . . . . . . . . . . . . . . . . . . . . 16.79. Plethysm corresponds to composition of operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.80. Calculation of plethysms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.84. Distributivity of the functorial plethysm operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Necklace rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.1. Necklace coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3. Necklace polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6. Dimension formula for the free associative algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.7. Necklace polynomial formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.11. The cyclotomic identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.13. Motivational remarks regarding the necklace algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.15. Deﬁnition of the necklace algebra functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.19. Binomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.22. Free binomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.24. Binomial rings vs λ-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.26. Frobenius and Verschiebung on necklace rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.29. Witt vectors vs necklaces in general . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.35. Modiﬁed necklace rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.38. Adjoints of the inclusion BinRing ⊂ CRing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.39. The functors BinW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.40. Carryless Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.41. To ponder and muse about . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.42. Apology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. Symm versus ⊕ R(Sn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 420 421 422 422 422 422 423 423 424 424 424 425 426 427 427 427 428 428 429 429 429 430 430 431 431 432 432 433 433 433 434 434 434

The ring R(S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hall inner product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Isometry theorem for the charateristic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hopf algebra R(S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frobenius identity (projection formula) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hopf algebras continue their insidious work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PSH algebras and the Zelevinsky classiﬁcation theorem . . . . . . . . . . . . . . . . . . . . . . . . . Bernstein morphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Majority ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjugate partition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gale-Ryser theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Snapper Liebler-Vitale Lam theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ruch-Schönhofer theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outer plethysm and inner plethysm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outer plethysm problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inner plethysm problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

434 435 435 436 437 437 437 438 439 440 440 441 441 441 441 442 442

18.1. 18.3. 18.6. 18.11. 18.14. 18.17. 18.19. 18.20. 18.24. 18.28. 18.30. 18.32. 18.33. 18.34. 18.35. 18.36. 18.37.

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19. Burnside rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.1. G-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4. Induction and restriction for G-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7. Almost ﬁnite G-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.10. Burnside theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.11. Burnside ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.12. Almost ﬁnite cyclic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ 19.13. A remarkable commutative diagram: the W (Z), B(Z), (Z), Nr(Z) isomorphisms . . . . . 19.16. Ghost component morphism for almost ﬁnite cyclic sets . . . . . . . . . . . . . . . . . . . . . . . . ˆ 19.21. The isomorphism T : W (Z) −→ B(Z) .................................... 19.31. Symmetric powers of G-sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ˆ 19.39. Frobenius and Verschiebung on W (Z), B(Z), (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.41. Burnside ring of a proﬁnite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.43. Existence theorem of the Witt-Burnside functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.46. β-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix on symmetric functions in an inﬁnity of indeterminates . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Power series in inﬁnitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.7. Symmetric and quasisymmetric power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10. Symmetric function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.11. Projective limit description of symmetric functions in an inﬁnity of variables . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

442 443 443 444 444 444 445 445 446 446 448 449 450 450 451 451 452 452 453 454 454 454

Part 2 of this survey chapter will include discussions of the following topics: Symm vs BU; Symm vs Exp; Symm vs Rrat (GL∞ ); Formal groups; Classiﬁcation of afﬁne commutative algebraic groups. Cartier duality; Covectors and bivectors; The K-theory of endomorphisms; de Rham-Witt complex. Witt vector cohomology; The lifting game; Witt vectors and deformations; Ramiﬁed Witt vectors; Noncommutative Witt vectors; The elementary particle view of Symm; Brief comments on more results on and applications of Witt vectors.

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1. Introduction and delimitation Let CRing be the category of commutative rings with unit element. Most of this chapter is about the big (or large or generalized) Witt vectors; that is about a certain functor W: CRing −→ CRing

(1.1)

that (therefore) assigns to each unital commutative ring A a new unital commutative ring W(A) (and to a unit element preserving ring morphism A −→ B a unital ring morphism W (A) −→ W(B)). It is also about the many quotient functors of W of which the most important are the p-adic Witt vectors Wp∞ , the truncated big Witt vectors Wn , and the truncated p-adic Witt vectors Wpn . Instead of ‘truncated’ one also ﬁnds ‘of ﬁnite length’ in the literature. The p-adic Witt vectors functor gets deﬁned via the socalled p-adic Witt polynomials p

Y0 , Y0 +pY1 ,

p2

p

Y0 +pY1 +p2 Y2 ,

p3

p2

p

Y0 +pY1 +p2 Y2 +p3 Y3 , · · · (1.2)

and their generalizations, the big Witt polynomials w1 (X) = X1 w2 (X) = X12 + 2X2 ···

··· n/d wn (X) = dXd ···

(1.3)

d|n

···

can2 be used to deﬁne the big Witt vectors. To see that (1.3) specializes to (1.2) relabel Xpi in wpn (X) as Yi . Here and there in the published literature one ﬁnds the Witt polynomials referred to as “mysterious polynomials that come out of nowhere.” It is one of my aims in the present screed to try to argue that that is simply not true. There is something inevitable about the Witt polynomials and they turn up naturally in various contexts and with some frequency. 1.4. Historically, this is also how things started. The setting is that of the investigations by Helmut Hasse and his students and collaborators into the structure of complete discrete valuation rings A with residue ﬁeld k. Oswald Teichmüller discovered that in such a situation there is a multiplicative system of representatives, i.e. a multiplicative section of the natural projection A −→ k. Such a system is unique if k is perfect.3 These are now called Teichmüller representatives. Assuming that A 2 But there are other ways. One of these will be used below (most of the time). 3 If k is not perfect a multiplicative system of representatives still exists but is not unique.

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is unramiﬁed, every element of A can be written as a power series in T , a generator of the maximal ideal of A, with coefﬁcients from any chosen system of representatives. Given the nice properties of the Teichmüller system it was and is natural to use these. As they are already multiplicative the ﬁrst problem was of course to ﬁgure out how Teichmüller representatives should be added in the arithmetic of the ring A. In the equal characteristic case, char(A) = char(k) = p > 0 that is no problem at all. The Teichmüller system is then additive as well. But in the unequal characteristic case, char(A) = 0, char(k) = p > 0 things are very different. In this case one can and does choose T = p. In 1936 Teichmüller [382], found a formula for doing precisely this, i.e. adding Teichmüller representatives. The formula is, see loc. cit. p. 156 a+b=

∞

cn pn

(1.5)

n=0

Here a and b are Teichmüller representatives of, say, a, b, and the cn are the Teichmüller representatives of elements cn in k that satisfy n

cpn = rn (a, b)

(1.6)

where the rn (X, Y) are the integer valued polynomials (recursively) determined by n

n−1

r0 (X, Y )p + pr1 (X, Y )p

n

n

+ · · · + pn rn (X, Y ) = Xp + Y p

(1.7)

And here they surface, on the left hand side of (1.7), the p-adic Witt polynomials. 1.8. Some 30+ years later, in another context, a very similar picture emerged. As a functor the big Witt vectors are isomorphic to the functor that assigns to a commutative unital ring A the Abelian group (A) of power series with constant term 1 and coefﬁents in A (under multiplication of power series). Much of Witt vector theory can be developed from this point of view without ever mentioning the generalized Witt polynomials (1.3). Except that it is far from obvious how to get the p-adic Witt vectors in this picture. In the main, this power series treatment is what is given below. But the Witt vectors themselves and the Witt polynomials will not be denied, as will be seen now. For any functor one is (or at least should be) interested in the operations on it, i.e. its functorial endomorphisms. In the present case a number of obvious and easy operations are the homothety operators uf(t) = f(ut) and the Verschiebung operators Vn f(t) = f(t n ). Rather less obvious are the Frobenius operators fn which are as much like raising to the n-th power as a morphism of Abelian groups can be. Now what is the sum (pointwise sum of operations) of two homothety operators? The answer is u + v =

∞ n=1

Vnrn(u, v)fn

(1.9)

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where this time the polynomials rn (X, Y ) are the integer valued polynomials determined recursively by drd (X, Y )n/d = Xn + Y n (1.10) d|n

and there they are, the generalized Witt polynomials (1.3).4 To appreciate the near perfect ﬁt of the picture of formulas (1.5)–(1.7) with that of (1.9)–(1.10), reﬂect that in a ﬁeld of characteristic p raising to the p-th power is the Frobenius morphism and that multiplication by p (in a characteristic zero ring, say, the p-adic integers) has a shift built into it. Further, the u, which are multiplicative, can be seen, in a very real sense, as Teichmüller representatives. 1.11. The underlying set of W(A) or, better (for the moment), (A), is the set of power series over A with constant term 1 (the unit element of A): (A) = {1 + a1 t + a2 t 2 + a3 t 3 + · · · : ai ∈ A for all i ∈ N}

(1.12)

(where, as usual, N denotes the natural numbers N = {1, 2, 3, · · · }). This functor (to Set, the category of sets), is obviously representable by the ring Symm = Z[h1 , h2 , h3 , · · · ]

(1.13)

of polynomials in an inﬁnity of indeterminates over the integers Z. As will rapidly become clear one really should see the hi as the complete symmetric functions (polynomials)5 in an inﬁnity of indeterminates6 ξ1 , ξ2 , ξ3 , · · · . Whence the notation used in (1.13). Also it turns out that many constructions, results, . . . for the Witt vectors have their natural counterparts in the theory of symmetric functions (as should be). Thus a chapter on the Witt vectors could very properly include most of symmetric function theory, that very large subject that could ﬁll several volumes. This will, of course, not be done. Also, there will be a separate chapter on the symmetric functions in this Handbook of Algebra. 1.14. Symm, the Hopf algebra of the symmetric functions is a truly amazing and rich object. It turns up everywhere and carries more extra structure than one would believe possible. For instance it turns up as the homology of the classifying space BU and also as the the cohomology of that space, illustrating its selfduality. It turns up as the direct sum of the representation spaces of the symmetric group and as 4 But in this case, I believe, the generalized Witt polynomials were known before formula (1.8) was written down. 5 Of course Symm is also equal to the polynomials in the elementary symmetric functions in the same inﬁnity of indeterminates and one could also work with those (and that is often done). But using the complete symmetric functions works out just a bit more elegantly, especially in connection with the autoduality of the Hopf algebra Symm. See sections 11.37 and 12. 6 Some of the readers hoped for may not be familiar with working with symmetric polynomials in an inﬁnity of indeterminates. There is really nothing to it. But for those that do not feel comfortable about it (and for the pernickety) there is a short appendix on the matter.

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the ring of rational representations of the inﬁnite general linear group. This time it is Schur duality that is involved. It is the free λ-ring on one generator. It has a nondegenerate inner product which makes it selfdual and the associated orthonormal basis of the Schur symmetric functions is such that coproduct and product are positive with respect to these basis functions. In this setting, positive, self dual, Hopf algebras with distinguished basis, there is a uniqueness theorem due to Andrei Zelevinsky [425]. But this is not yet entirely satisfactory unless it can be shown that the Schur symmetric functions are in some algebraic way canonical (which seems very likely). Symm is also the representing ring of the functor of the big Witt vectors and the covariant bialgebra of the formal group of the big Witt vectors (another manisfestation of its autoduality). Most of these things will be at least touched on below. As the free λ-ring on one generator it of course carries a λ-ring structure. In addition it carries ring endomorphisms which deﬁne a functorial λ-ring structure on the rings W(A) = CRing(Symm, A) for all unital commutative rings A. A sort of higher λring structure. Being selfdual there are also co-λ-ring structures and higher co-λ-ring structures (whatever those may be). Of course, Symm carries still more structure: it has a second multiplication and a second comultiplication (dual to each other) that make it a coring object in the category of algebras and, dually, (almost) a ring object in the category of coalgebras. The functor represented by Symm, i.e. the big Witt vector functor, has a comonad structure and the associated coalgebras are precisely the λ-rings. All this by no means exhausts the manifestations of and structures carried by Symm. It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one, and many more uniqueness theorems are needed. In this chapter I will only touch upon the aspects of Symm which relate directly to the Witt vector constructions and their properties. To conclude this introduction let me remark that the Witt vector construction is a very beautiful one. But is takes one out of the more traditional realms of algebra very quickly. For instance the ring of p-adic Witt vectors of a ﬁeld is Noetherian if and only if that ﬁeld is perfect, [63], Chapter IX, page 43, exercise 9. Also, much related, the p-adicWitt vector ring Wp∞ (k[T ]) of one dimensional afﬁne space over a characteristic p > 0 ﬁeld is not Noetherian, which would appear to rule out any kind of systematic use of the Witt vectors in algebraic geometry as currently practiced.7 It is perhaps because of this that the Witt vector functors and the rings and algebras thus arising have not really been much studied,8 except in so far as needed for their applications (which are many and varied). 1.15. Initial sources of information. There are a number of (electronic) encyclopedia articles on the Witt vectors, mostly the p-adic Witt vectors, see [4, 5, 6, 7]. These can 7 This is not really true; witness crystalline cohomologyy and the de Rham-Witt complex (of which cristalline cohomology is the hyper homology) which is made up of W(k) modules. 8 Should a reader inadvertently get really interested in the Witt vector ring functors he/she is recommended to work through the 57 exercises on the subject in [63], 26 pp worth for the statements only, mostly contributed, I have been told, by Pierre Cartier.

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serve to obtain a ﬁrst sketchy impression. There are also a number of (introductory) lecture notes and chapters in books on the subject, see e.g. [57, 74, 86, 188], [192 Chapter 3], [212, 263], [271, §26] [326 Lectures 6–12], [336]. However, the full story of the Witt vectors is a long and varied one. Here, in the present chapter. I try to present a ﬁrst outline. 2. Terminology The big and p-adic and truncated Witt vectors carry ring and algebra structures, and hence, naturally, are sometimes referred to as a ring or algebra of Witt vectors and then (by erosion) as Witt ring and Witt algebra. This is a bit unfortunate and potentially confusing because these phrases mostly carry other totally unrelated meanings. Mostly, ‘Witt ring’ refers to a ring of equivalence classes of quadratic forms (with addition and multiplication induced by direct product and tensor product respectively), see e.g. [96, 296]. Further ‘Witt algebra’ mostly refers to something like the Lie algebra of differd ential operators spanned by the di = xi+1 dx , i ∈ Z under the commutator product [di , dj ] = (j − i)di+j , i.e. the centerless Virasoro algebra, much studied in physics; see e.g. [182, 410, 426]. The fact that the sub Lie algebra spanned by the di for i ≥ −1 is usually denoted W1 adds a bit more potential confusion, and this becomes worse when one encounters Wn for the more variable version of this algebra of differential operators. On the other hand the term ‘Witt group’ mostly indeed refers to a group of Witt vectors.9 3. The p-adic Witt vectors. More historical motivation It has become customary to ‘motivate’ the introduction of the Witt vectors by looking at unramiﬁed complete discrete valuation ﬁelds, or, more precisely their rings of integers. To start with, consider the ring of p-adic integers Zp , i.e. the completion of the integers with respect to the norm ||n|| = p−vp (n) where the valuation vp (n) of an integer n is the largest power of p that divides n. Here p is a prime number. Every p-adic integer α, i.e. element of Zp , can be uniquely written as a convergent sum α = a0 + a1 p + a2 p2 + a3 p3 + · · · , ai ∈ {0, 1, · · · , p − 1}

(3.1)

More generally, instead of the set {0, 1, · · · , p − 1} one can choose any set of representatives of the residue ﬁeld Z/(p) = Fp = GF(p).10 Now what happens if two such expressions are added or multiplied; i.e. what are the coefﬁcients of a sum or product of p-adic integers? As in the case of the familiar decimal notation for arithmetic 9 But not always, cf e.g. [82], which is again about quadratic form related matters. 10 To use three of the standard notations for the prime ﬁeld of p elements.

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with integers this involves carry-overs. That is perfectly acceptable for a calculating machine but irritating to algebraists who would like universal formulas which always work. Things get much worse if the residue ﬁeld is not a prime ﬁeld where it is not even clear (a priori) what set of representatives to choose. Of course at this stage it is far from obvious whether a universal formula can exist; that even seems unlikely at ﬁrst sight. As it turns out there are universal formulas: the addition and multiplication polynomials of the p-adic Witt vectors. However, judging from the introduction in Witt’s seminal paper [420] the arithmetic of p-adic ﬁelds was not Witt’s primary motivation. This seems also indicated by the fact that this arithmetic is not mentioned in the title of the paper, but only in the subtitle. Instead Witt seems to have been mainly motivated by a desire to obtain a theory of cyclic Galois extensions for general charateristic p ﬁelds similar to the theory he had himself obtained for power series ﬁelds in [415, 416] and, especially, to understand some mysterious formulas of Hermann Ludwig Schmid, [355] concerning central cyclic algebras in characteristic p. All this more or less in the framework of a class ﬁeld theory for function ﬁelds that was being vigorously developed at the time by H Hasse, F K Schmidt, O Teichmüller, H L Schmid, C Chevalley, E Witt and others. See [338] for a thorough and very readable account of this part of the history of class ﬁeld theory. Some of the important original papers are [16, 23, 87, 191, 190, 355, 356, 357, 382, 415, 416, 420]. The situation as regards these cyclic central algebras was as follows. For α, β ∈ k (of characteristic p > 0) let (α, β] denote the simple central cyclic11 algebra of degree p with two generators u, θ subject to the relations up = α,

θ p − θ = β,

uθu−1 = θ + 1

In the Brauer group there are the relations (α, β] · (α , β] = (αα , β]

and (α, β] · (α, β ] = (α, β + β ]

H L Schmid in [355] succeeded in deﬁning simple central cyclic algebras of degree pn denoted (α|β] = (α|β0 , β1 , . . . , βn−1 ] and by dint of some heroic formula manipulation found rules of the form (α|β] · (α |β] = (αα |β] and (α|β] · (α|β ] = (α|s0 (β, β ), · · · , sn−1 (β, β )] with, as Witt writes, “certain polynomials si (x, y) that are initially deﬁned in characteristic 0 and only after they have been proved to be integral are taken modulo p”. In his fundamental paper [420] Witt proves that these formulas are in fact the addition formulas of the p-adic Witt vectors; see also section 5 below. I will also brieﬂy return to these algebras in section 8 below. 11 The word ‘cyclic’ here refers to the fact that the algebras considered are cross products involving a

cyclic Galois group; see [96], chapter 7, especially §7.2 and §7.5 for more detail. See also section 8 below.

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So, here is already a third way in which the Witt vector polynomials turn up naturally and unavoidably. 4. Teichmüller representatives Let A be a ring with an ideal m such that the quotient ring k = A/m of characteristic p > 0 is perfect, which means that the ring morphism σ: k −→ k, x → xp , the Frobenius morphism, is supposed to be bijective.12 Suppose, moreover, that A is complete in the m-adic topology. For instance A can be the ring of p-adic numbers Zp , the quotient ring Z/(pm ), or the ring of power series k((T )) over a perfect ﬁeld k of characteristic p. Here the corresponding ideals m are respectively the principal ideals pZ, pZ/(pm ), and (T ). There is now the following simple observation: For all a, b ∈ A, if a ≡ b mod mr , r ≥ 1, then ap ≡ bp mod mr+1 (4.1) Now for any x ∈ k take lifts yr ∈ A of σ −r (x), r = 0, 1, 2, · · · ; i.e. q(yr ) = pr where q: A −→ k is the canonical projection, so that all the yr are lifts of

σ −r (x)

p

p2

pr

x. Now consider the sequence y0 , y1 , y2 , · · · , yr , · · · . It follows from (4.1) and the completeness hypothesis on A that this sequence converges to a limit that is a lift of x. Moreover, again by (4.1), this limit does not depend on the choice of the yr . This limit is denoted t(x) or tA (x) and called the Teichmüller representative of x. This Teichmüller system of representatives has the following properties t(0) = 0, t(1) = 1, t(xx ) = t(x)t(x )

(4.2)

i.e. it is multiplicative. (And if A is also of characteristic p it is also additive: tA (x + x ) = tA (x) + tA (x ). But in general this is not the case.) The Teichmüller system is also the unique multiplicative one and the unique one which commutes with p-th powers. 5. Construction of the functor of the p-adic Witt vectors Let p be a prime number and consider the following polynomials in a countable inﬁnity of commuting indeterminates X0 , X1 , X2 , · · · w0 (X) = X0 p w1 (X) = X0 + pX1 ··· ··· pn X0

wn (X) = ··· ···

pn−1 + pX1

p + · · · + pn−1 Xn−1

(5.1) + pn Xn

12 For instance the ﬁnite ﬁelds GF(pf ) are perfect. There are also perfect rings that are not ﬁelds, for −i instance, the ring k[T p , i = 0, 1, 2, · · · ] for a perfect ﬁeld of constants k.

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These are called the (p-adic) Witt polynomials. Now there occurs what has been called ‘the miracle of the Witt polynomials’. This is the following integrality statement. 5.2. Theorem. Let ϕ(X, Y, Z) be a polynomial over the integers in three (or less, or more) commuting indeterminates. Then there are unique polynomials ϕn (X0 , X1 , · · · , Xn ; Y0 , Y1 , · · · , Yn ; Z0 , Z1 , · · · , Zn ) = ϕn (X; Y ; Z), n = 0, 1, 2, · · · such that for all n wn (ϕ0 (X; Y ; Z), · · · , ϕn (X; Y ; Z)) = ϕ(wn (X), wn (Y ), wn (Z))

(5.3)

The proof is really quite simple. For instance by induction using the following simple observation. 5.4. Lemma. Let ψ(X) = ψ(Xi , i ∈ I ) be a polynomial (or power series for that matter) in any set of commuting indeterminates with integer coefﬁcients. Write ψ(Xp ) for the polynomial obtained from ψ(X) by replacing each indeterminate by its p-th power. Then ψ(Xp ) ≡ ψ(X)p mod p

j

j+1

and ψ(Xp )p ≡ ψ(X)p

mod pj+1

(5.5)

Note the similarity with (4.1). 5.6. Proof of theorem 5.2. Obviously the polynomials ϕn are unique and can be recursively calculated from (5.3) over the rationals, starting with ϕ0 (X0 ; Y0 ; Z0 ) = ϕ(X0 ; Y0 ; Z0 ). This also provides the start of the induction. So suppose with induction that the ϕi have been proved integral for i = 0, 1, · · · , n − 1. Now observe that wn (X) ≡ wn−1 (Xp ) mod p and hence ϕ(wn (X), wn (Y ), wn (Z)) ≡ ϕ(wn−1 (Xp ), wn−1 (Y p ), wn−1 (Zp ))

= wn−1 (ϕ0 (Xp ; Y p ; Zp ), · · · , ϕn−1 (Xp ; Y p ; Zp )) (5.7)

Using (5.5) one has pi

pi−1

pn−i ϕn−i (X; Y ; Z) ≡ pn−i ϕn−i (Xp ; Y p ; Zp ) mod pn and so the n terms of the last expression in (5.7) are term for term equal modpn to the ﬁrst n terms of pn

pn−1

wn (ϕ0 (X; Y ; Z), · · · , ϕn (X; Y ; Z)) = ϕ0 +pϕ1

p

+· · ·+pn−1 ϕn−1 +pn ϕn .

Hence pn ϕn (X; Y ; Z) ≡ 0 modpn , i.e. ϕn (X; Y ; Z) is integral. There are also other proofs; for instance a very elegant one due to Lazard, [258], and reproduced in [367]. 5.8. It is obvious from the proof that the theorem holds for polynomials ϕ in any number of variables or power series in any number of variables, but things tend to get a bit messy notationally. It is also obvious that there are all kinds of versions for other

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rings of coefﬁcients; see also subsection 9.77–9.98 below for some more remarks on this theme. Actually, as will be seen later, see section 9 on the big Witt vectors, theorem 5.2 is not at all necessary for the construction of the various functors of Witt vectors and to work with them; nor, for that matter the Witt polynomials. There are several other ways of doing things. 5.9. The p-adic Witt addition and multiplication polynomials. The addition polynomials sn (X0 , · · · , Xn ; Y0 , · · · , Yn ) and multiplication polynomials mn (X0 , · · · , Xn ; Y0 , · · · , Yn ) of the p-adic Witt vectors are now deﬁned by wn (s0 , s1 , · · · , sn ) = wn (X) + wn (Y ) and wn (m0 , m1 , · · · , mn ) = wn (X)wn (Y ) 5.11. And the functor of the p-adic Witt vectors itself is deﬁned as Wp∞ (A) = (a0 , a1 , · · · an , · · ·) : ai ∈ A = AN∪{0}

(5.10)

(5.12)

as a set and with multiplication and addition deﬁned by a +W b = (a0 , a1 , · · · , an , · · · ) +W (b0 , b1 , · · · , bn , · · · ) = (s0 (a;b), s1 (a;b), · · · , sn (a;b0 ), · · · ) a ·W b = (a0 , a1 , · · · , an , · · · ) ·W (b0 , b1 , · · · , bn , · · · )

(5.13)

= (m0 (a;b), m1 (a;b), · · · , mn (a;b0 ), · · · ) There is a zero element, viz (0, 0, 0, · · · ) and a unit element, viz (1, 0, 0, · · · ) and the claim is: 5.14. Theorem. The sets Wp∞ (A) together with the addition and multiplication deﬁned by (5.9) and the unit and zero element as speciﬁed deﬁne a commutative unital ring valued functor, where the ring morphism corresponding to a ring morphism α: A −→ B is component wise, i.e. Wp∞ (α)(a0 , a1 , · · · ) = (α(a0 ), α(a1 ), · · · ). Moreover, the Witt polynomials wn deﬁne functor morphisms wn : Wp∞ (A) −→ A, a = (a0 , a1 , a2 , · · · ) → wn (a). Finally for Q-algebras w: Wp∞ (A) −→ AN∪{0} , a → w(a) = (w0 (a), w1 (a), · · · ) is an isomorphism onto the ring AN∪{0} with componentwise addition and multiplication. Wp∞ is also (obviously) the unique functor which setwise looks like A → AN∪{0} and for which the wn are functorial ring morphisms. The elements wn (a) for a Witt vector a are often called the ghost components of that Witt vector (originally: ‘Nebenkomponente’). 5.15. Proofs of theorem 5.14. To prove commutativity, associativity, distributivity and that the zero and unit element have the required properties there are several methods. One is to use deﬁning polynomials as in (5.3). For instance the sequences of polynomials s(m(X;Y ); m(X;Z)) and m(X;s(Y ;Z)) both satisfy (5.3) for the

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polynomial ϕ(X, Y, Z) = X(Y + Z) = XY + XZ and so they are equal proving distributivity on the left. Another way is to rely on functoriality. First, by the last line in the theorem (which is obvious) distributivity etc. hold for Q-algebras. Then, because a commutative integral domain embeds injectively into its ring of quotients, the required properties hold for integral domains. Finally for every unital commutative ring there is an integral domain that surjects onto it and so it follows that the required properties hold for every unital commutative ring. 5.16. Ghost component equations. Universal calculations (= ‘Calculating with universal polynomials’). The polynomials sn (X;Y ), mn (X; Y ) are solutions of what I call ‘ghost component equations’. For instance the ghost component equations for addition can be written wn (addition) = wn (X) + wn (Y ),

n = 0, 1, 2, · · ·

(5.17)

and call for a sequence of polynomials s(X;Y ) = (s0 (X;Y ), s1 (X;Y ), · · · ) such that wn (s) = wn (X) + wn (Y ). For the unit Witt vector one has the ghost component equations wn (unitvector) = 1,

n = 0, 1, 2, · · ·

(5.18)

which call for a series of polynomials u (which turn out to be constants) such that wn (u) = 1 for all n. (So that u0 = 1, un = 0 for n ≥ 1.) There will be many ‘ghost component equations’ below. They constitute a most useful and elegant tool, though, as has already been remarked on, one can perfectly well do without them. Not all of these ghost component equations fall within the scope of theorem 5.2, see e.g. subsections 5.25 and 5.27 below. Here are some more general properties of the p-adic Witt vector functor. 5.19. Ideals and topology. For each n = 1, 2, 3, · · · mn = {(0, · · · , 0, an , an+1 , an+2 , · · · ): aj ∈ A}

(5.20)

is a (functorial) ideal in Wp∞ (A); further mi mj ⊂ mmax(i,j) (obviously, and no more can be expected in general;13 for instance the multiplication polynomial m1 has the term pX1 Y1 and so in general m1 m1 ⊂ m2 ). The ring Wp∞ (A) is complete and Hausdorff in the topology deﬁned by these ideals. The quotients Wpn (A) = Wp∞ (A)/mn+1 are the rings of p-adic Witt vectors of ﬁnite length n + 1. 5.21. Teichmüller representatives. For each x ∈ A let t(x) = tWp∞ (A) (x) = (x, 0, 0, · · · ). This is the Teichmüller representative of x (for the natural projection w0 : Wp∞ (A) −→ A = Wp∞ (A)/m1 ). This system of representatives is indeed multiplicative because, as is easily checked, m0 (X0 ; Y0 ) = X0 Y0 and mn (X0 , 0, · · · , 0; Y0 , 0, · · · , 0) = 0 for n ≥ 1. 13 But things change very much for the case of p-adic Witt vectors over a ring of characteristic p; see section 6.

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5.22. Multiplication with a Teichmüller representative. It is an easy exercise to check that for the multiplication polynomials pn

mn (X0 , 0, · · · , 0; Y0 , Y1 , · · · , Yn ) = X0 Yn

(5.23)

and so in every Wp∞ (A) p

pn

(a0 , 0, · · · ) ·W (b0 , b1 , · · · , bn , · · · ) = (a0 b0 , a0 b1 , · · · , a0 bn , · · · ) (5.24) 5.25. Verschiebung. Consider the ghost component equations w0 (Vp ) = 0, wn (Vp ) = pwn−1

for n ≥ 1

(5.26)

These call for a series of polynomials v = (v0 , v1 , v2 , · · · ) such that w0 (v) = 0, wn (v) = pwn−1 (v) for n ≥ 1 and do not fall within the scope of theorem 5.2. The immediate and obvious solution is v0 (X) = 0, vn (X) = Xn−1

for n ≥ 1

and so for each a = (a0 , a1 , · · · ) ∈ Wp∞ (A) the operation Vp acts like Vp a = (0, a0 , a1 , a2 , · · · ). Then because wn (Vp a) = pwn−1 (a), wn (Vp (a +W b)) = wn (Vp a) + wn (Vp b), and it follows that Vp deﬁnes a functorial group endomorphism of the Witt vectors.14 It does not respect the multiplication. It is called Verschiebung. Note that, see (5.19) mi (A) = Vpi (Wp∞ (A)) 5.27. Frobenius. The ghost component equations for the Frobenius operation are wn (fp ) = wn+1 ,

n = 0, 1, 2, · · ·

(5.28)

This calls for a sequence of polynomials f = (f0 , f1 , f2 , · · · ) such that wn (f ) = wn+1 , a set of equations that also falls outside the scope of theorem 5.2. For one thing the polynomial fn involves the indeterminate Xn+1 ; for instance p

p

f0 = X0 + pX1 , f1 = X1 + pX2 −

p−1

pp−i−1

i=0

p ip p−i p X0 X1 ≡ X1 modp i (5.29)

It is easy to show that the fn are integral and that, moreover, fn (X) ≡ Xnp modp,

n = 0, 1, 2, 3, · · ·

(5.30)

14 There are pitfalls in calculating with ghost components as is done here. Such a calculation gives a valid proof of an identity or something else only if it is a universal calculation; that is, makes no use of any properties beyond those that follow from the axioms for a unital commutative ring only. That is the case here. See also the second proof of theorem 5.14 in 5.15.

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so that for a ring k of characteristic p the Frobenius operations on the p-adic Witt vectors over k are given by p

p

p

fp (a0 , a1 , a2 , · · · ) = (a0 , a1 , a2 , · · · )

(5.31)

It follows from the ghost component equations (5.28) that the Frobenius operation fp (a0 , a1 , a2 , · · · ) = (f0 (a), f1 (a), f2 (a), · · · )

(5.32)

is a functorial endomorphism of unital rings of the functor of the p-adic Witt vectors. 5.33. Multiplication by p for the p-adic Witt vectors. The operation of taking p-fold sums on the p-adic Witt vectors, i.e. a → p ·W a = a +W a +W · · · +W a is (of course) given by the polynomials Pn = s(s(· · · (s(X; X); X); · · · ; X) which satisfy the ghost component equations wn (P0 , P1 , P2 , · · · ) = pwn (X)

(5.34) p

p2

The ﬁrst two polynomials are P0 = pX0 , P1 = X0 + pX1 − pp−1 X0 . With induction one sees p

Pn (X) ≡ Xn−1 mod p

for n ≥ 1

(and P0 (X) ≡ 0 mod p)

(5.35)

For a change, here are the details. According to (5.34) the polynomials Pn are recursively given by the formulas pn

pn−1

P0 + pP1

p

+ · · · + pn−1 Pn−1 + pn Pn

pn

pn−1

= pX0 + p2 X1

p

+ · · · + pn Xn−1 + pn+1 Xn

(5.36) pn−i

p

With induction, assume Pi ≡ Xi−1 modp for i ≥ 1. This gives Pi pn−i

pn−i+1

pn−i+1

≡ Xi−1

modpn−i+1 and pi Pi ≡ pi Xi−1 modpn+1 . So the middle n − 1 terms of the left hand side of (5.36) are term for term congruent to the ﬁrst n − 1 terms of the right hand side leaving pn

p

P0 + pn Pn ≡ pn Xn−1 + pn+1 Xn modpn+1 p

so that indeed Pn ≡ Xn−1 modp because P0 = pX0 and pn > n. 5.37. Taking p-th powers in the Witt vectors. This operation is governed by the ghost component equations wn (M0 , M1 , M2 , · · · ) = wn (X)p

(5.38)

These polynomials Mi are of course integral as they can be obtained by repeated substitution of the p-adic Witt vector multiplication polynomials into themselves. For the ﬁrst few polynomials Mi one ﬁnds (by direct calculation) p

p−1

M0 (X) = X0 , M1 (X) ≡ pX0

X1 modp2 ,

p2 −1

M2 (X) ≡ X0

p

X1 modp (5.39)

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5.40. Adding ‘disjoint’ Witt vectors. Finally, suppose for two p-adic Witt vectors a = (a0 , a1 , a2 , · · · ) and b = (b0 , b1 , b2 , · · · ) it is the case that for every n at least j

pj

pj

one of an or bn is zero. Then (ai + bi )p = ai + bi for each i, j and it follows that wn (a0 + b0 , a1 + b1 , a2 + b2 , · · · ) = wn (a) + wn (b) so that in such a case a +W b = (a0 + b0 , a1 + b1 , a2 + b2 , · · · ). More generally it now follows that each Witt vector a = (a0 , a1 , a2 , · · · ) is equal to the unique convergent sum a=

∞ i=0

Vpi (t(ai ))

(5.41)

5.42. Product formula. The Frobenius operator, Verschiebung operator, and multiplication for Witt vectors are related in various ways. One of these is the product formula, [192], page 126, formula (17.3.17): Vp (a · (fp b)) = Vp (a) · b

(5.43)

This is a kind of formula that shows up in various parts of mathematics, such as when dealing with direct and inverse images of sheaves in algebraic geometry, in (algebraic) K-theory and in (abstract representation theory when dealing with restriction and induction of representations.15 For instance in representation theory such a formula holds with ‘induction’ in place of Vp and ‘restriction’ in place of f p . It is important enough in representation theory that it has become an axiom in the part of abstract representation theory known as the theory of Green functors. There it takes the form, [409], page 809: H H (a · RH IK K (b)) = IK (a) · b

Sometimes this axiom is called the Frobenius axiom. Further in algebraic geometry one has a natural isomorphism f∗ (F ⊗OX f ∗ E) ∼ = f∗ F ⊗OX E for a morphism of ringed spaces f: (X, OX ) −→ (Y, OY ) and suitable sheaves E and F where f∗ and f ∗ stand for taking direct and inverse images under f , see [189], exercise 5.1(d) on page 124, while in (étale) cohomology one ﬁnds in [295], Chapter VI, §6, proposition 6.5, page 250, a cupproduct formula i∗ (i∗ (x) ∪ y) = x ∪ i∗ (y). These types of formulae are variously called both product formulae and projection formulae. It is absolutely not an accident that the same kind of formula turns up in Witt vector theory. To prove the product formula (5.43) one does a universal calculation. Apply the ghost component morphism wn to both sides of (5.43) to ﬁnd respectively (for n ≥ 1): wn (Vp (a · fp b) = pwn−1 (a · fp b) = pwn−1 (a)wn−1 (fp b) = pwn−1 (a)wn (b) wn (Vp a · b) = wn (Vp a)wn (b) = pwn−1 (a)wn (b) 15 The terminology ‘product formula’ is not particularly fortunate. There are many things called ‘product

formula’ in many part of mathematics. Mostly they refer to formulae that assert that an object associated to a product is the product of the objects associated to the constituents or to such formulae where a global object for e.g. a global ﬁeld is the product of corresponding local objects. This happens e.g. for norm residue symbols in class ﬁeld theory. The terminology ‘projection formula’ (which is also used for formulas like (5.43) is also not particularly fortunate and suffers from similar defects.

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and these are equal. As also applying w0 to the two sides gives the same result, the product formula is proved. 5.44. Here is another interrelation between the Frobenius and Verschiebung operators and multiplication: fp Vp = [p]

(5.45)

where [p] stands for the operation that takes a p-adic Witt vector into the p-fold sum of itself, a → a + W a + W · · · + W a. But in general Vp fp is not equal to [p]. To prove (5.45) one again does a universal calculation: wn (fp Vp a) = wn+1 (Vp a) = pwn (a) and also wn ([p]a) = pwn (a). 6. The ring of p-adic Witt vectors over a perfect ring of characteristic p In this section k will ﬁnally be a perfect ring of characteristic p > 0. In this case the ring of p-adic Witt vectors over k has a number of additional nice properties including a ﬁrst nice universality property. Recall that ‘k is perfect’ means that the Frobenius ring morphism σ: k −→ k, x → xp is bijective. One of the aims of this section to show that if k is a perfect ﬁeld of characteristic p then the p-adic Witt vectors form a characteristic zero complete discrete valuation ring with residue ﬁeld k. 6.1. p-adic Witt vectors over a ring of characteristic p. First, just assume only that k is of characteristic p. Then calculations with p-adic Witt vectors simplify quite a bit. For instance: p

p

p

fp (a0 , a1 , a2 , · · · ) = (a0 , a1 , a2 , · · · ) p

p

p

[p](a0 , a1 , a2 , · · · ) = p · (a0 , a1 , a2 , · · · ) = (0, a0 , a1 , a2 , · · · ) p = (0, 1, 0, 0, 0, · · · )

(6.2)

fp Vp = [p] = Vp fp pj pi

Vpi a · Vpj b = Vpi+j (a0 b0 , ??, ??, · · · ) where the ??’s in the last line stand for some not speciﬁed polynomial expressions in the coordinates (components) of a and b. The ﬁrst four of these formulas follow directly from (5.25), (5.30) and (5.35). The last one follows from the fourth one, repeated application of the product formula, and the ﬁrst formula of (6.2) (and commutativity of the multiplication). Note that the last formula implies that mi mj ⊂ mi+j

(6.3)

(but equality need not hold). Also there is the corrollary 6.4. Corollary. If k is an integral domain of characteristic p then the ring of p-adic Witt vectors over k is an integral domain of characteristic zero.

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6.5. Valuation rings. A (normalized) discrete valuation16 on a (commutative) ﬁeld K is a surjective function v: K −→ Z ∪ {∞} such that v(x) = ∞

if and only if

x = 0,

v(xy) = v(x) + v(y),

(6.6)

v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. Such a valuation can be used to deﬁne a norm, and hence a topology, on K by setting ||x|| = r −v(x) where r is e.g. an integer > 1. This deﬁnes an ultrametric or non-Archimedean metric, d(x, y) = ||x − y||, on K where ‘ultra’ means that instead of the familiar triangle inequality one has the stronger statement d(x, y) ≤ max{d(x, z), d(y, z)}. The valuation ring of the valued ﬁeld (K, v) is the ring A = {x ∈ K: v(x) ≥ 0}

(6.7)

Note that such a ring automatically has two properties A is local with maximal ideal m = {x ∈ K: v(x) ≥ 1}

(6.8)

The maximal ideal m is principal

(6.9)

One can also start with a valuation on an integral domain A, which now is a surjective function v: A −→ N ∪ {0, ∞} such that the properties (6.6) hold. A valuation on the ﬁeld of fractions Q(A) is then deﬁned by v(xy−1 ) = v(x) − v(y) (which is independent of how an element of the ﬁeld of fractions is written as a fraction). Such an integral domain with a valuation is not necessarily a valuation ring. In particular, one or both of the properties (6.8), (6.9) may fail. 6.10. Example. The p-adic valuation on the integers. Take a prime number p. Deﬁne on the integers the function vp (n) = r if and only if pr is the largest power of the prime number p that divides n. This is obviously a valuation, but Z is not its corresponding valuation ring. That valuation ring is the localization Z(p) consisting of all rational numbers that can be written as fractions with a denominator that is prime to p. 6.11. However, if A is an integral domain with a valuation and the ring A satisﬁes properties (6.8), (6.9), then A is a discrete valuation ring. See e.g. [136], page 50, theorem 7.7; [62], Chapter 5, §3, no 6, proposition 9, page 100. After this intermezzo on discrete valuation ringss let’s us return to rings of p-adic Witt vectors. 16 Such valuations as here are also sometimes called ‘exponential valuations’. More generally one considers such functions with values in the real numbers (or more general ordered groups) with inﬁnity adjoined, cf e.g. [136], page 20. Then the value group v(K\{0}) is a subgroup of the additive group of real numbers. Such a group is either discrete (when there is a smallest positive real number in it) or dense. In the discrete case one may as well assume that the value subgroup is the group of integers. Whence the terminology ‘normalized discrete’.

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6.12. Units in the ring of p-adic Witt vectors. Let k be any ring of characteristic p, then a p-adic Witt vector a = (a0 , a1 , a2 , · · · ) is invertible in Wp∞ (A) if and only if a0 is invertible in k. Indeed, multiplying with (a0−1 , 0, 0, 0, · · · ) it can be assumed by (5.24) that a0 = 1 (multiplication with a Teichmüller representative). Then a = 1 + Vp b with b = (a1 , a2 , · · · ). The series 1 − Vp b + (Vp b)2 − (Vp b)3 + · · · converges in Wp∞ (k) because of (6.2) or (6.3), say, to an element c. Then a · c = 1. It follows in particular that if k is a ﬁeld of characteristic p then Wp∞ (k) is a local ring with maximal ideal m = m1 = Vp Wp∞ (k). 6.13. Rings of p-adic Witt vectors over a perfect ring of characteristic p. Now let k be a ring of characteristic p that is perfect. There are of course such rings that are not p ﬁelds, for instance a ring k [T1 , T2 , · · · ] with the relations Ti = Ti−1 for i ≥ 2 and −1 −2 k a perfect ﬁeld. This ﬁeld can be suggestively written k [T, T p , T p , · · · ]. Now ﬁrst note that if k is perfect of characteristic p then the ideal m = m1 = Vp Wp∞ (k) is principal and generated by the element p ∈ Wp∞ (k). This is immediate from the second formula in (6.2) above. Second, observe that in this case mr = pr Wp∞ (k) = (m1 )r = Vpr Wp∞ (k). In this setting there is the ﬁrst nice universality property of the p-adic Witt vectors. 6.14. Theorem. Let k be a ring of characteristic p that is perfect. Let A be a ring with an ideal a such that A/a = k and such that A is complete and separated in the topology deﬁned by a. Then there is a unique morphism of rings ϕ such that the following diagram commutes Wp∞ (k) ↓ w0 k

ϕ

−→ =

A ↓q k

Here the topology deﬁned by a is the topology deﬁned by the sequence of ideals an ,

n = 1, 2, · · · and ‘separated’ means that an = {0}. Further q is the canonical quon

tient mapping A −→ A/a = k. It is perfectly OK for A to be a ring of characteristic p or for it to be such that ar = 0 for some r. 6.15. Proof of theorem. 6.14. First, let’s prove uniqueness of ϕ (if it exists at all). To this end observe that by the remarks in (6.13) and (5.41) every element in Wp∞ (k) can be uniquely written in the form of a convergent sum ∞ n=0

−n

t(anp )pn ,

an ∈ k

and thus (by the completeness and separatedness of A) a ring morphism ϕ : Wp∞ (k) −→ A is uniquely determined by what it does to the Teichmüller representives. But, as ϕ is a ring morphsim and hence multiplicative, the ϕ(t(a)), a ∈ k from a system of Teichmüller representatives for k in A. As such systems are unique it follows that there can be at most one ϕ that does the job.

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For existence consider the ghost component ring morphism wn: Wpn (A) = Wp∞ (A)/mn+1 (A) −→ A If r0 , r1 , · · · , rn ∈ a then wn (r1 , r2 , · · · , rn ) ∈ an+1 and so there is an induced ring morphism ψn that makes the following diagram commutative wn

Wpn (A) ↓ Wpn (q)

−→

Wpn (k)

−→ A/an+1

A ↓

ψn

Now deﬁne ϕn: Wpn (k) −→ A/an+1 as the composite Wpn (k)

Wpn (σ −n )

−→

ψn

Wpn (k) −→ A/an+1

Then for any a0 , a1 , · · · , an ∈ A it follows that pn

pn

pn

n

pn−1

ϕn (q(a0 , q(a1 ), · · · , q(anp )) = a0 + pa1

+ · · · + pn an

(6.16)

Now let x0 , x1 , · · · , xn ∈ k and substitute in this formula (6.16) the Teichmüller p−n representatives (in A) tA (xi ) to ﬁnd that −1 −2 p p + p 2 t A x2 + ··· ϕn (x0 , x1 , · · · , xn ) = tA (x0 ) + ptA x1 −n

+ pn tA (xnp ) mod an+1 and so the projective limit ϕ of the ϕn exists, is a ring morphism, and is given by the formula p−1

p−2

ϕ(x0 , x1 , x2 , · · · ) = tA (x0 ) + ptA (x1 ) + p2 tA (x2 ) + · · ·

(6.17)

Of course one can write down this formula directly but then one is faced with proving that it is a ring morphism which comes down to much the same calculations as were carried out just now. Now suppose in addition that a = (p) and that A is of characteristic zero. Then the ring morphism ϕ

Wp∞ (k) −→ A, (x0 , x1 , x2 , · · · ) → x0 + ptA (x1 )p + p2 tA (x2 )p + · · · (6.18) 2

is clearly bijective and hence an isomorphism. So there is a (strong) uniqueness result as follows 6.19. Theorem. Let k be a perfect ring of characteristic p > 0. Then, up to isomorphism, there is precisely one ring A of characteristic zero with residue ring A/(p) = k and complete in the p-adic topology. Moreover A is rigid in the sense that the only ring automorphism of A that induces the identity on k is the identity on A.

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6.20. The p-adic Witt vectors over a perfect ﬁeld of characteristic p > 0. Finally let k be a perfect ﬁeld of characteristic p. Then by 6.13, m = m1 = Vp Wp∞ (k) is generated by p. Deﬁne v as follows v : Wp∞ (k) −→ N ∪ {0, ∞}, (6.21) v(0) = ∞, v(x0 , x1 , x2 , · · · ) = n if xn is the ﬁrst coordinate unequal to zero Then v is a valuation on the ring of p-adic Witt vectors and makes this ring into a complete discrete valuation ring by (6.11) and (6.12). And this ring is unramiﬁed as the maximal ideal is generated by p. Theorem 6.19 now translates into an existence and uniqueness (and rigidity) theorem for complete discrete unramiﬁed valuation rings of characteristic zero and residue characteristic p > 0. Thus, for a perfect residue ﬁeld there are indeed universal formulae that govern the addition and multiplication of p-adic expansions as in section 3 above. The treatment here has a bit of a ‘deus ex machina’ ﬂavor in that the p-adic Witt vectors are constructed ﬁrst and subsequently proved to do the job. Motivationally speaking one can do better. Once it is accepted that it is a good idea to write p-adic expansions using Teichmüller representatives one can calculate and come up with addition and multiplication formulae. This is nicely done in [188]. It was also already known in 1936, [382], that there should be integral coefﬁcient formulae that could do the job.17 Witt’s major contribution was ﬁnding a way to describe them nicely and recursively. 6.22. Complete discrete valuation rings with nonperfect residue ﬁeld. Immediately after Witt’s paper, in the same18 issue of ‘Crelle’, there is a paper by Teichmüller, [380], in which he proves existence and uniqueness of unramiﬁed complete discrete valuation rings with given residue ﬁeld, thus completing his own arguments from [382]. For the unequal characteristic case he uses the p-adic Witt vectors. Existence and uniqueness of unramiﬁed complete discrete valuation rings had been treated before by Friedrich Carl Schmidt and Helmut Hasse in [191], but that paper had an error, which was later, in 1940, pointed out by Saunders MacLane, [277], and still later corrected by him [278]. For more see [339]. 6.23. Cohen rings. The unique characteristic zero complete unramiﬁed discrete valuation ring with a given residue ﬁeld k of characteristic p > 0 is nowadays often 17 Indeed in [382], the theorem at the bottom of page 58 says the following. Deﬁne polynomials n n pn pn−1 p h0 , h1 , h2 , · · · in two variables by the formulae xp + yp = h0 + ph1 + · · · + pn−1 hn−1 + pn hn . Then these have integer coefﬁcients. Given a, b in the perfect residue ﬁeld k, let cn , n = 0, 1, 2, · · · be pn given by cn = hn (a, b). Then the p-adic expansion of the sum to the Teichmüller representatives of a and b is t(a) + t(b) = t(c0 ) + pt(c1 ) + p2 t(c2 ) + · · · , which of course nicely agrees with the addition of

p-adic Witt vectors via the isomorphism (6.17). Thus the addition formula for Teichmüller representatives was known, and given by universal formulae, and involved what were to be called Witt polynomials. The multiplication of Teichmüller representatives was of course also known (as these are multiplicative). Distributivity then determines things recursively. There was still much to be done and no prediction that things would come out as nicely as they did, but the seeds were there. 18 The two papers carry the same date of receipt.

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called the Cohen ring19 of k; see [63] Chapter IX, §2 for a treatment of Cohen rings.20 If k is perfect the Cohen ring of k is the ring of p-adic Witt vectors. If k is not perfect it is a ‘much smaller’ ring. The technical deﬁnition of Cohen ring in loc. cit. is as follows. A p-ring is (by deﬁnition) a unital commutative ring such that the ideal pC is maximal and the ring is complete and separated in the p-adic topology deﬁned by the powers of this ideal. Given a local separated complete ring A a Cohen subring of it is a p-ring that satisﬁes A = mA + C (where mA is the maximal ideal of A. The existence and uniqueness of such a subring of the p-adic Witt vectors is implicit in the work of Teichmüller, [380] and Nagata, [304], Chapter V, §31, p. 106ff, but a nice functorial description had to wait till the work of Colette Schoeller, [358].21 6.24. p-basis. Let k be a ﬁeld of characteristic p > 0. The ﬁeld is perfect if and only if k = kp = {xp : x ∈ k}. If it is not perfect there exists a set of elements B = {bi : i ∈ I} of k such that the monomials

j bi i , ji ∈ {0, 1, · · · , p − 1}, ji equal to zero for all but ﬁnitely many i i∈I

form a basis (as a vector space) for k over the subﬁeld kp . The cardinality of I, which can be anything, is an invariant of k and sometimes called the imperfection degree of k. Such a set is called a p-basis. The notion is again due to Teichmüller and ﬁrst appeared in [381] (for other purposes than valuation theory). For some theory of p-bases see [64], Chapter 5, §8, exercise 1, and [304], p. 107ff. If B is a p-basis for k the monomials

j bi i , ji ∈ {0, 1, · · · , pn − 1}, ji equal to zero for all but ﬁnitely many i i∈I

n

form a vector space basis for k over the subﬁeld kp . 6.25. Cohen functor. Given a characteristic p > 0 ﬁeld k and a chosen p-basis let Z[B] be the ring of polynomials over the integers in the symbols from B. Then Schoeller deﬁnes functors Cn on the category of commutative unital Z[B]-algebras 19 There are other rings that are called ‘Cohen rings’ in the published literature. First there are certain twisted power series rings which are a kind of noncommutative complete discrete valuation rings, see e.g. [131, 342, 406]. Further in [137], section 18.34B, p. 48, a Cohen ring is deﬁned as a ring R such that R/P is right Artinian for each nonzero prime ideal P; see also [138, 413]. This last bit of terminology has to do with the paper [93]. 20 This terminology would appear to come from the so-called Cohen structure theorems for complete local rings, [94]. 21 Existence of a multiplicative system of representatives is again crucial. In the general case, i.e., when the residue ﬁeld k is not necessarily perfect there is still existence (as noticed by Teichmüller, [382]), but no uniqueness. A very nice way of seeing existence is due to Kaplansky, [235] section 26, p. 84ff, who notes that the group of 1-units (Einseinheiten) of A, i.e. the invertible elements that are mapped to 1 under the residue mapping, form a direct summand of the group of all invertible elements of A. This remark seems to have been of some importance for Schoeller, and reference 4 in her paper, a paper that never appeared, should probably be taken as referring to this.

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to the category of unital commutative rings. These functors depend on the choice of the p-basis. They deﬁne afﬁne group schemes. Now if the characteristic of the base ﬁeld k is p and perfect the ﬁnite length p-adic Witt vector group schemes serve to classify unipotent afﬁne algebraic groups over k, a topic which will get some attention in a later section of this chapter. For a non-perfect base ﬁeld the Cohen functors deﬁned in [358], see also [248] are a well-working substitute.22 6.26. Cohen ring of k. It follows from (6.25) that Fp [B], where Fp is the ﬁeld of p elements, naturally embeds in k; combined with the canonical projection Z[B] −→ FP [B] this deﬁnes a Z[B]-algebra structure on k. The value of the Cohen functors on k can now be described as follows, see [358], section 3.2, p. 260ff; see also [63], Chapter IX, §2, exercise 10, p. 72. n

Cn (k) is the subring of Wn+1 (k) generated by Wn+1 (kp ) ⊂ Wn+1 (k) and the Teichmüller representatives (b, 0, 0, · · · , 0) ∈ Wn+1 (k), b ∈ B These form a projective system (with surjective morphisms of rings Cn+1 (k) → Cn (k) induced by the projections Wn+1 (k) → Wn (k)) and taking the projective limit one ﬁnds a subring C(k) of the ring of p-adic Witt vectors which is a discrete complete unramiﬁed valuation ring. It is unique but one loses the rigidity property that holds for the Witt vectors, see theorem 6.19 and 6.20. 7. Cyclic Galois extension of degree pn over a ﬁeld of characteristic p As already indicated in the introduction (section 1) for Witt himself one of the most important aspects of the Witt vectors was that they could be used to extend and complete his own results from [416, 415] to obtain a Kummer theory (class ﬁeld theory) for Abelian extensions of a ﬁeld of characteristic p. This section is a brief outline of this theory. In this whole section p is a ﬁxed prime number. 7.1. Construction of some Abelian extensions of a ﬁeld of characteristic p > 0. Consider the functor of p-adic Witt vectors Wp ∞ . It is an easy observation that for each n ≥ 1, the multiplication and addition polynomials si (X; Y ), mi (X; Y ), i = 0, · · · , n − 1 only depend on X0 , · · · , Xn−1 ; Y0 , · · · , Yn−1 . It follows immediately that the sets Vpn Wp∞ (A) = {(0, · · · , 0, a0 , a1 , · · · , am , · · · )}

(7.2)

n

are (functorial) ideals in the rings Wp∞ (A). And hence, for each n, there is a quotient functor Wpn−1 (A) = Wp∞ (A)/Vpn Wp∞ (A) = {(a0 , a1 , · · · , an−1 ) : ai ∈ A}

(7.3)

22 There may be some difﬁculties with the constructions in [358] in case the cardinality of the p-basis is

not ﬁnite as is stated in [378], where also an alternative is proposed.

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These are called the p-adic Witt vectors of length n. The multiplication and addition of these vectors of length n are given by the multiplication and addition polynomials si (X; Y ), mi (X; Y ), i = 0. · · · , n − 1. In other words Wpn−1 is the functor on CRing to CRing represented by Z[X0 , X1 , · · · , Xn−1 ] provided with the coring object structure given by the polynomials si (X; Y ), mi (X; Y ), i = 0. · · · , n − 1. If the ring A is of characteristic p the Frobenius endomorphism on Wp∞ (A) is given by (see (6.2) above) p

p

p

fp (a0 , a1 , a2 , · · · ) = (a0 , a1 , a2 , · · · )

(7.4)

and thus manifestly takes the ideal Vpn Wp∞ (A) into itself. In general this is not the case. Thus for rings of characteristic p there is an induced (functorial) Frobenius endomorphism of rings fp : Wpn−1 (A) → Wpn−1 (A)

(7.5)

From now on in this section k is a ﬁeld of characteristic p > 0. Consider the additive operator p: Wpn−1 (k) → Wpn−1 (k), p(α) = fp (α) − α

(7.6)

(Witt vector subtraction, of course.) For n = 1, so that Wpn−1 (k) = k this is the wellknown Artin-Schreier operator xp − x which governs the theory of cyclic extensions of degree p of the ﬁeld k. For n > 1 the operator (7.6) is very similar, particularly if one reﬂects that the Frobenius operator is ‘as much like raising to the power p as an additive operator on the Witt vectors can be’ (as has been remarked before). Moreover, consider the kernel of the operator p. For a given n his kernel consists p of all p-adic Witt vectors α = (a0 , a1 , · · · , an−1 ) of length n such that ai = ai for all i = 0, 1, · · · , n − 1. Thus for all these i, ai ∈ Fp ⊂ k the prime subﬁeld of p elements of k. Thus Ker(p) = Wpn−1 (Fp ) = Z/pn Z

(7.7)

the cyclic group of order pn . (The last equality in (7.7) is immediate from the considerations of the previous chapter.) This is most encouraging in that it suggests that the study of equations fp (α) − α = p(α) = β

(7.8)

in Wpn−1 (k) with β ∈ Wpn−1 (k) and k an algebraic closure of k could lead to cyclic extensions of degree pn just like in standard Artin–Schreier theory as described e.g. in [96], pp. 205–206. This is indeed substantially the case and things turn out even better. For any β ∈ Wpn−1 (k) consider a solution α = (a0 , a1 , · · · , an−1 ) of (7.8). Then k(p−1 β) denotes the extension ﬁeld of k generated by the components

) is another solution a0 , a1 , · · · , an−1 . Because of (7.7) if α = (a0 , a1 , · · · , an−1

) so that the extension k(p−1 β) of k does k(a0 , a1 , · · · , an−1 ) = k(a0 , a1 , · · · , an−1

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not depend on the choice of a solution. More generally if is any subset of Wpn−1 (k) then k(p−1 ) denotes the union of all the k(p−1 β) for β ∈ . 7.9. The main theorems for Abelian extension of exponent pn of ﬁelds of characteristic p. There is now sufﬁcient notation to formulate the main theorems of Kummer theory for Abelian extensions of exponent pn of a ﬁeld of characteristic p. 7.10. Theorem (Kummer theory). Let ⊂ Wpn−1 (k) be a subgroup which contains pWpn−1 (k) and such that /pWpn−1 (k) is ﬁnite. Then the Galois group of the extension k(p−1 ) is isomorphic to /pWpn−1 (k). For each Abelian ﬁeld extension K/k of exponent a divisor of pn there is precisely one group /pWpn−1 (k) such that K = k(p−1 ). 7.11. Theorem (on cyclic extensions of degree pn ). Let β = (b0 , b1 , · · · , bn−1 ) be / pWpn−1 (k). Then k(p−1 β) is a a p-adic Witt vector of length n over k such that b0 ∈ n cyclic extension of degree p of k. A generator of the Galois group is given by σ(α) = α + 1 (Witt vector addition, and 1 the unit of the ring Wpn−1 (k)). All cyclic extensions of degree pn can be obtained in this way. 7.12. On the proofs. The theorems above already occur in [420]. But the proofs there are very terse. They consists of brief instructions to the reader to ﬁrst prove a kind of (additve) “Hilbert 90 theorem” for Witt vectors by redoing the proof of theorem 1.2 in [415]. This says that a ﬁrst Galois cohomology group with coefﬁcients in Witt vectors is zero and is ‘Satz 11’ in [420]. The further instructions are to redo the arguments of [415] using vectors instead of numbers and using ‘Satz 11’ instead of the usual ‘Hilbert 90’ as the occasions demand. For a good complete treatment of this Kummer theory for Abelian extensions of ﬁelds of characteristic p > 0 see [271], pp. 146–150. The statements there are slightly more general and a bit more elegant than in [420] in that the group /pWpn−1 (k) is not required to be ﬁnite. The isomorphism statement of theorem 7.10 now becomes a duality statement to the effect that the group /pWpn−1 (k) and the Galois group of the extension are dual to each other under a natural pairing. 8. Cyclic central simple algebras of degree pn over a ﬁeld of characteristic p The second main application of the p-adic Witt vectors in [420] is to cyclic central simple algebras of prime power degree pn over a ﬁeld of characteristic p > 0. This is a topic in the theory of simple central algebras over and the Brauer group of a ﬁeld, [1, 2], [96] Chapter 7, [102] Chapter III. 8.1. Central simple algebras. A central simple algebra over a ﬁeld k is what the name indicates: it is a ﬁnite dimensional algebra over k, it is simple, i.e. no nontrivial ideals, and it is central, i.e. its centre, {a ∈ A: ar = ra for all r ∈ A} coincides with k. A

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central division algebra over k is a central simple algebra in which every nonzero element is invertible. The classic example is the algebra of quaternions over the reals. For every central simple algebra A there is a (unique up to isomorphism) central division algebra D such that A is isomorphic to a full matrix ring over D. For every central simple algebra over k there is a ﬁeld extension23 K/k such the tensor product Ak = A ⊗k K is a full matrix ring over K. Such a ﬁeld is called a splitting ﬁeld. The tensor product (over k) of two central simple algebras over k is again a central simple algebra over k. 8.2. Brauer group. Two central simple algebras A, B, over k are called (Brauer) equivalent if for suitable natural numbers m and n A ⊗k Mm (k) ∼ = B ⊗k Mn (k)

(8.3)

Here Mm (k) is the full matrix algebra over k of all m × m matrices with entries in k. Equivalently, A and B are equivalent if they have isomorphic associated division algebras. The tensor product is compatible with this equivalence notion and deﬁnes (hence) a group structure on the equivalence classes. This group is called the Brauer group, Br(k), of k. It can be interpreted as a second Galois cohomology group. 8.4. Crossed product central simple algebras, [102] §III-2, [96] §7.5. A central simple algebra over k is called a cross product if it contains a maximal subﬁeld K such that K/k is a Galois extension. That K is then a splitting ﬁeld. This is the ‘abstract’ deﬁnition of a crossed product. There is also an explicit description/construction which is important and explains the terminology. This goes as follows. Let U be the group of invertible elements of a crossed product central simple algebra A. Let N be the centralizer of K× = K\{0} in U: N = {u ∈ U: u−1 Ku ⊆ K} Then K×

(8.5)

is a normal subgroup of N and hence gives rise to a short exact sequence

of groups 1 −→ K× −→ N −→ −→ 1

(8.6)

a group extension of by K× . One easily shows that ∼ = Gal(K/k). Being a group extension, it is determined by what is called a factor system and this same factor system can be used to recover (up to isomorphism) the crossed product central simple algebra A. Not every central simple algebra is a crossed product, but every Brauer equivalence class contains one. 23 As always the term ‘ﬁeld’ implies commutativity.

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M. Hazewinkel

When the Galois extension involved is cyclic, i.e. the group = Gal(K/k) is cyclic, one speaks of a cyclic central simple algebra. 8.7. Cyclic central algebra associated to a ﬁnite length p-adic Witt vector. From now on in this section k is a ﬁxed ﬁeld of characteristic p > 0. Take an element α ∈ k and a ﬁnite length Witt vector β = (b0 , b1 , · · · , bn−1 ) ∈ Wpn−1 (k). To these data associate the algebra generated over k by an indeterminate u and commuting indeterminates θ0 , θ1 , · · · , θn−1 subject to the relations n

up = α,

pθ = β,

uθu−1 = θ + 1

(8.8)

Here, θ = (θ0 , θ1 , · · · , θn−1 ) (as a Witt vector) and uθu−1 = (uθ0 u−1 , uθ1 u−1 , · · · , uθn−1 u−1 ). This algebra will be denoted (α|β]. These algebras are central simple algebras of ‘degree’ pn (meaning that their dimension over k is p2n ). Note that conjugation by u is an operation of order pn . Also, by the results of Section 7 above, if b0 ∈ / pWpn−1 (k) the subalgebra k(θ0 , θ1 , · · · , θn−1 ) is a cyclic ﬁeld extension of degree pn and conjugation by u is the action of a generator of the Galois group. This is just about sufﬁcient to prove that (α|β] is a cyclic central simple (crossed product type) algebra. In case b0 ∈ pWpn−1 (k) there is a ‘reduction theorem’, Satz 15 from [420], which says the following. 8.9. Theorem. The algebra (α|0, b1 , b2 , · · · , bn−1 ] is Brauer equivalent to (α|b1 , b2 , · · · , bn−1 ] The next theorem is rather remarkable and shows once more that there is no escaping the Witt vectors. 8.10. Theorem (Brauer group and Witt vectors). In the Brauer group of the ﬁeld k there are the (calculating) rules (α|β] · (α |β] = (αα |β] (α|β] · (α|β ] = (α|β + β ]

(8.11)

(where in the second line of (8.11) the ‘+’ sign means Witt vector addition. For the time being the sections above conclude the discussions on p-adic Witt vectors. There will be more about the p-adic Witt vectors in various sections below. But ﬁrst it is time to say something about that truly universal object the functor of the big Witt vectors; and that will be the subject in the next few sections. 9. The functor of the big Witt vectors In the early 1960’s, probably independent of each other, several people, notably Ernst Witt himself and Pierre Cartier, noticed that the p-adic Witt polynomials (5.1) are but part of a more general family and that these polynomials can be used in a similar manner to deﬁne a ring valued functor W : CRing −→ CRing of which the p-adic Witt vectors are a quotient, see [105, 149, 192, 255, 419]. This functor W is called

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349

the functor of the big Witt vectors.24 (And also, over Z(p) -algebras the canonical projection W(A) −→ Wp∞ (A) has an Abelian group section, making in this case the p-adic Witt vectors also a sub functor of the big Witt vectors.) Here, I will ﬁrst construct the big Witt vectors in another way, before describing those ‘generalized’ Witt polynomials alluded to, and how the p-adic Witt vectors ﬁt with the big ones. 9.1. The functor of one power series. For each ring (unital and commutative) let (A) = 1 + tA[[t]] = {f = f(t) = 1 + a1 t + a2 t 2 + · · · : ai ∈ A}

(9.2)

be the set of power series over A with constant term 1. Under the usual multiplication of power series this is an Abelian group (for which the power series 1 acts as the zero element). Thus (9.2) deﬁnes an Abelian group valued functor : CRing −→ AbGroup. The morphism of Abelian groups associated to a morphism α: A −→ B is coefﬁcient-wise, i.e. (α)(1 + a1 t + a2 t 2 + a3 t 3 + · · · ) = 1 + α(a1 )t + α(a2 )t 2 + α(a3 )t 3 + · · · (9.3) Giving a power series like (9.2) is of course the same thing as giving an inﬁnite length vector (a1 , a2 , a3 , · · · ) and in turn such a vector is the same as a morphism of commutative rings ϕf

Symm = Z[h1 , h2 , h3 , · · · ] = Z[h] −→ A, ϕf : hi → ai

(9.4)

where, as indicated in the notation, Symm is the ring of polynomials in an inﬁnity of commuting indeterminates h1 , h2 , h3 , · · · . Thus the functor is representable by Symm. The functorial addition on the Abelian group (A) then deﬁnes a comultiplication on Symm μS = μSum : Symm −→ Symm ⊗ Symm, hn → hi ⊗ hj (9.5) i+j=n

where by deﬁnition and for ease of notation h0 = 1. This makes Symm a Hopf algebra. The comultiplication formula of course encodes the ‘universal’ formula (that is ‘recipe’) for multiplying power series which is (1 + a1 + a2 + a3 + · · · )(1 + b1 + b2 + b3 + · · · ) = 1 + c1 + c2 + c3 + · · · ⇔ cn = ai bj i+j=n

with, again, a0 = b0 = c0 = 1. Which, in turn, is the same as saying that the addition on CRing(Symm, A) is given by the convolution product μs

ϕf ⊗ϕg

mA

ϕfg = Symm −→ Symm ⊗ Symm −−−−→ A ⊗ A −−→ A where mA is the multiplication on the ring A. 24 In earlier writings also ‘generalized Witt vectors’.

(9.6)

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M. Hazewinkel

As in the case of the p-adic Witt vectors of the previous 7 sections it is often convenient (but never, strictly speaking, necessary) to work with ghost components. These are deﬁned by the formula s(f ) = s(a1 , a2 , a3 , · · · ) = s1 t + s2 t 2 + s3 t 3 + · · · = t

tf (t) d log(f(t)) = dt f(t)

(9.7)

so that for example the ﬁrst three ghost components are given by the universal formulas s1 (a) = −a1 , s2 (a) = a12 − a2 , s3 (a) = −a13 + 2a1 a2 − a3

(9.8)

Because of the properties of ‘log’ as deﬁned formally by x2 x3 x4 xn + − + · · · + (−1)n+1 + · · · (9.9) 2 3 4 n so that log(fg) = log(f ) + log(g), this implies that addition in (A) (which is multiplication of power series) corresponds to coordinate-wise addition of ghost components ( just as in the case of the p-adic Witt vectors). log(1 + x) = x −

9.10. The symmetric function point of view. Now imagine that the ai are really the complete symmetric functions in a set of elements ξ1 , ξ2 , ξ3 , · · · (living as it were in some larger ring containing A); i.e. an = hn (ξ1 , · · · , ξn , · · · )

(9.11)

where the hn (X) are the familiar complete symmetric functions in the commuting indeterminates X1 , X2 , X3 , · · · , viz Xj1 Xj2 · · · Xjn (9.12) hn (X) = j1 ≤j2 ≤···≤jn

The relation (9.11) can be conveniently written down in power series terms as ∞

i=1

1 = 1 + a1 t + a2 t 2 + a3 t 3 + · · · = f(t) = f (1 − ξi )

(9.13)

The ghost components of f are given by (9.7). In terms of the ξi that means that d d s1 t + s2 t 2 + s3 t 3 + · · · = t log f(t) = t log (1 − ξi t)−1 dt dt i

ξ2 t2 ξ3 t 3 d −ξi t − i − i − · · · = −t dt 2 3 =

∞

(9.14)

i

j j j ξ1 + ξ 2 + ξ 3 + · · · t j

j=1

so that in terms of the ξ’s the ghost components are given by the power sums. 9.15. Multiplication on the Abelian groups (A). If there is to be a ring multiplication on (A) then in particular there should be a multiplication of the very special power series (1 − xt)−1 and (1 − yt)−1 . Just about the simplest thing one can imagine

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351

is that the product of these very special power series is (1 − xt)−1 ∗ (1 − yt)−1 = (1 − xyt)−1

(9.15)

Something which ﬁts with what has been seen in the theory of the p-adic Witt vectors: the (1 − xt)−1 are the Teichmüller representatives of the x ∈ A which should be multiplicative. Distributivity and functoriality together now force that the multiplication of power series in (A) should be given by

f(t) = (1 − ξi t)−1 , g(t) = (1 − ηi t)−1 i

i

⇒ (f ∗ g)(t) = (1 − ξi ηj t)−1

(9.16)

i,j

Note that this formula makes perfectly good sense. The right most expression is symmetric in the ξ’s and also symmetric in the η’s and so, when written out, gives a power series with coefﬁcients that are complete symmetric functions in the ξ’s and in the η’s. And that means that they are universal polynomials in the coefﬁcients of f and g. Here is how this multiplication works out on the ghost components. As in (9.14) d (1 − ξi ηj t)−1 = ((ξi ηj )t + (ξi ηj )2 t 2 + (ξi ηj )3 t 3 + · · · ) t log dt i,j

i,j

=

∞

(9.17) pn (ξ)pn (η)t n

n=1

Thus, multiplication according to (9.16) translates into component-wise multiplication for the ghost components. Actually because of distributivity it would have been sufﬁcient to do this calculation for (9.15). Associativity of the multiplication and distributivity follow by functoriality (or directly for that matter). All in all, what has been deﬁned is a unital-commutative-ring-valued functor : CRing −→ CRing

(9.18)

together with a set of ‘ghost component’ functorial ring morphisms sn : (A) −→ A

(9.19)

which determine the ring structure on (A) by the very requirement that they be functorial ring morphisms. The unit element of the ring (A) is the power series 1 + t + t 2 + t 3 + · · · = (1 − t)−1 . 9.20. The functor of the big Witt vectors. Now what has all this to do with Witt-vector-like constructions? As a set deﬁne W(A) = AN = {(x1 , x2 , x3 , · · · ): xi ∈ A}

(9.21)

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and set up a functorial bijection (of sets) between W(A) and (A) by eA : (x1 , x2 , x3 , · · · ) →

∞

(1 − xn t n )−1

(9.22)

n=1

and transfer the functorial ring structure on the (A) to the W(A) by this bijection. This then deﬁnes a unital-commutative-ring-valued functor W : CRing −→ CRing (that is isomorphic to , so one might justiﬁably wonder why one takes the trouble). Let’s calculate the ghost components of a Witt vector (x1 , x2 , x3 , · · · ) ∈ W(A), which by deﬁnition means calculating wn (x) = wn (x1 , x2 , x3 , · · · ) = sn (eA (x))

(9.23)

This is easy t

d d log (1 − xd t d )−1 = t − −m−1 (xd t d )m dt dt m d

d

=

d,m

dxdm t dm

=

∞ n=1 d|n

n/d

dxd

(9.24)

and thus the ghost components of the ring valued functor W are given by the (generalized) Witt polynomials n/d wn (X1 , X2 , · · · , Xn ) = dXd (9.25) d|n

Note that these do indeed generalize the Witt polynomials (5.1) of the previous 8 sections in that, apart from a relabeling the polynomials wpn of (9.25) are indeed the polynomials wn of (5.1). From now on in this whole chapter, that is in all sections that follow, wn and wn (X) and · · · refer to the polynomials as deﬁned by (9.25). Incidentally, the formula for (second) multiplication of power series, i.e. multiplication in the ring (A) of power series written in ‘x-coordinates’ as in the right hand side of (9.22) can be written down directly as follows d

(1 − xd t d )−1 ∗ (1 − yn t n )−1 = (1 − xrm/r ysm/s t m )−rs/m (9.27) n

r,s≥1

where on the right hand side m is the least common multiple of r and s. This is a formula that was known to Witt, [419]. Thus there is no absolute need to introduce symmetric functions into the game; it is just very convenient. As far as I know the ﬁrst substantial treatment of the big Witt vectors from the symmetric functions point of view is due to Pierre Cartier, [84, 86].

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9.28. The ai of (9.2) and (9.13), and the xi of (9.22) can be best seen as simply being different ways to coordinatize (A). As is usual with two different coordinate systems each has its advantages and diadvantages. One good property of the ‘x-coordinates’ has already been pointed out. They show that the ring (A) is a (far reaching) generalization of the p-adic Witt vectors. Another is that the ghost component formulae, i.e. the Witt polynomials wn (X) are very simple. For one thing they do not involve mixed terms in the Xi ; for another wn (X) depends only on those Xd for which d is a divisor of n. This latter fact makes it clear that there are many interesting quotient functors of W (which is difﬁcult to see in the formulation). These quotient functors will be described in the next section, 10. On the other hand there are good well known formuals both ways (from symmetric function theory) that relate the ‘a-coordinates’ (i.e. the power series coordinates) with their ghost components while inverting the Witt polynomials seems to be a bit of a mess. Quite a number of formulas from symmetric function theory are important (or at least very useful) for Witt vector theory. In the next subsections there are a few. 9.29. Very partial symmetric function formularium (1).25 Everything takes place in the rings Z[ξi : i ∈ I] respectively

Z[[ξi : i ∈ I]]

of polynomials, respectively power series, in a countable inﬁnite collection of commuting indeterminates over the integers. (And also in their analogues Q[ξi : i ∈ I] and Q[[ξi : i ∈ I]] over the rationals.) For some deﬁnitional and foundational remarks on what it means to have a polynomial or power series in an inﬁnite number of variables see the appendix to this chapter. The rings just above are graded by giving all the ξi degree 1. The reader who does not like polynomials and power series in an inﬁnite number of variables can just imagine that things are in terms of a ﬁnite number of them, large enough for the business at hand. 9.30. Partitions and monomial symmetric functions. A partition λ = (λ1 , λ2 , · · · , λn ) is a ﬁnite sequence of elements from N ∪ {0} in non-increasing order, λ1 ≥ λ2 ≥ · · · ≥ λi ≥ · · · . Partitions with different numbers of trailing zeros are identiﬁed. The length of a partition, lg(λ), is the number of its non zero entries; its weight is wt(λ) = λ1 + λ2 + · · · + λn . A different notation for a partition is (1m1 (λ) 2m2 (λ) · · · nmn (λ) ) where mi (λ) is the number of entries of it that are equal to i. A partition of weight n is said to be a partition of n and its entries are (also) called the parts of that partition. For any sequence α = (a1 , a2 , · · · , am ) of elements from N ∪ {0} let am ξ α = ξ1a1 ξ2a2 · · · ξm , e.g. ξ (0,0,0,2,0,3,0,0) = ξ42 ξ63

(9.31)

25 Not all of the formulas and functions that follow belong to the standard formularium of the symmetric functions.

354

M. Hazewinkel

Given a partition λ (strictly speaking with an inﬁnite string of trailing zeros added), the associated monomial symmetric function is the sum mλ = ξα (9.32) α

where the sum is over all distinct sequences α (inﬁnite with only a ﬁnite number of nonzero entries) that are permutations of λ (with trailing zeros added). Thus, for example, m(2,1) =

i=j

m(1,1) =

ξi2 ξj = ξ12 ξ2 + ξ22 ξ1 + ξ12 ξ3 + ξ32 ξ1 + ξ22 ξ3 + ξ32 ξ2 + · · · ξi ξj = ξ1 ξ2 + ξ1 ξ3 + ξ2 ξ3 + · · ·

(9.33) (9.34)26

i<j

These are the simplest symmetric functions in that if a symmetric function contains one of the monomials from an mλ then it also contains all others (with the same coefﬁcient). The monomial symmetric function mλ is homogeneous of degree wt(λ) = n and they form a basis for the free Abelian group of symmetric functions of weight n. By deﬁnition the complete symmetric functions and the elementary symmetric functions are mλ and en = m(1,1,··· ,1) (n1’s) (9.35) hn = wt(λ)=n

and for a partition λ = (λ1 , λ2 , · · · , λm ) one writes hλ = hλ1 hλ2 · · · hλm

and eλ = eλ1 eλ2 · · · eλm

(9.36)27

The complete symmetric functions and the elementary ones are related by the Wronski relations (−1)i hi ej = 0 for n ≥1, where h0 = e0 = 1 (9.37) i+j=n

The hλ (resp. eλ ) for λ of weight n also form a basis for the symmetric functions of degree n. By deﬁnition Symm = ⊕n Symmn ⊂ Z[ξ]

(9.38)

is the graded ring of symmetric functions in the commuting indeterminates ξ. In this whole chapter it will (usually) be seen as the graded ring in the complete 26 Note the difference between the two formulae which rests on the word ‘distinct’ in the sentence just below (9.32). 27 Note that this notation has a rather different meaning than that for m . It is deﬁnitely not true that m λ λ is equal to something like m(λ1 ) m(λ2 ) . . . m(λm ) .

Witt vectors. Part 1

355

symmetric functions Symm = Z[h1 , h2 , · · · , hm , · · · ] = Z[h]

(9.39)

For many purposes one could equally well work with the elementary symmetric functions, but the complete ones just work out better. Actually, there are precisely four ‘canonical’ choices, see subsection 10.18 below on the Liulevicius theorem. 9.40. Inner product. One deﬁnes a (symmetric positive deﬁnite) inner product on Symm by declaring the monomial and the complete symmetric functions to form a bi-orthogonal system: < hλ , mκ > = δλ,κ (Kronecker delta)

(9.41)28

There is now a remarkable theorem, see [281], (4.6), p. 63: 9.42. Theorem. Let {uλ } and {vλ } be two sets of symmetric functions indexed by all partitions (including the zero partition) that both form a basis for SymmQ = Symm ⊗Z Q (or Symm itself ). Then they form a bi-orthonormal system (i.e < uλ , vλ >= δλ,κ ) if and only if

uλ (ξ)vλ (η) = (1 − ξi ηj )−1 (9.43) λ

i,j

Here η = {η1 , η2 , · · · } is a second set of commuting indeterminates that also commute with the ξ’s, and vλ (η) is the same as vλ (ξ) = vλ but now written as an expression in the η’s. The theorem is especially remarkable to a mathematician with (products of ) Witt vectors on his mind29 and I am far from sure that its consequences in that context have been fully explored and understood. Theorem 9.42 ﬁts with formula (9.41) for it is indeed the case that ([281], p. 62) λ

hλ (ξ)mλ (η) =

(1 − ξi ηj )−1

(9.44)

i,j

9.45. Schur functions. A symmetric function, i.e. an element of Symm, is called positive if when expressed as a sum in the monomial basis {mλ } all its coefﬁcient are non negative. All the speciﬁc symmetric functions introduced so far, the mλ , hλ , eλ , are positive. It now turns out that there is a unique positive orthonormal basis of Symm. They are called the Schur functions, and are determined by

sλ (ξ)sλ (η) = (1 − ξi ηj )−1 (9.46) λ

i,j

and the positivity requirement. 28 This inner product is often called ‘Hall inner product’. 29 See (9.16).

356

M. Hazewinkel

The ﬁrst few Schur functions are s(1) = m(1) = h1 ,

s(2) = m(2) + m(1,1) = h2 ,

s(1,1) = m(1,1) = −h2 + h(1,1) (9.47)

and the ones in weights 3 and 4 are given by the matrices ⎛ ⎞ ⎛ ⎞ 1 0 0 1 −1 1 ⎝1 1 0⎠, s-m matrix; ⎝0 1 −2⎠, s-h matrix 1 2 1 0 0 1 ⎛

1 ⎜1 ⎜ ⎜1 ⎜ ⎝1 1

0 1 1 2 3

0 0 1 1 2

0 0 0 1 3

⎞ 0 0⎟ ⎟ 0⎟ ⎟, s-m matrix; 0⎠ 1

⎛

1 −1 0 ⎜0 1 −1 ⎜ ⎜0 0 1 ⎜ ⎝0 0 0 0 0 0

1 −1 −1 1 0

(9.48)

⎞ −1 2⎟ ⎟ 1⎟ ⎟, s-h matrix −3⎠ 1 (9.49)

where the partitions have been ordered lexicographically, (4), (3,1), (2,2), (2,1,1), (1,1,1,1), and the columns give the coefﬁcients of the Schur functions in terms of the monomial symmetric functions (resp. the complete symmetric functions). Thus e.g. s(3,1) = m(3,1) + m(2,2) + 2m(2,1,1) + 3m(1,1,1,1) s(2,1,1) = h(4) − h(3,1) − h(2,2) + h(2,1,1)

(9.50)30

There are many ways to deﬁne the Schur functions and to write down formulas for them. The standard (and oldest) deﬁnition is the one by Jacobi, see [281], p. 40ff. An explicit expression is given by the Jacobi–Trudy formulas sλ = det(hi − i + j)1≤i,j≤n Other expressions come from the Schur identity

sλ = (1 − ξi )−1 (1 − ξi ξj )−1 λ

i

(9.51)

(9.52)

i<j

(see [273], 5.4, p. 176). This one has the advantage of showing immediately that Schur symmetric functions are positive (which is far from obvious from (9.51)). 30 Calculating with the monomial symmetric functions directly is a bit messy (and so is calculating with their explicit expressions). For instance m(3,1) m(1) = m(4,1) + m(3,2) + 2m(3,1,1) , m(4,1) m(2) = m(6,1) + m(4,3) + m(4,2,1) , m(2,1) m(1) = 2m(2,2) + m(3,1) + 2m(2,1,1) . It is fairly easy to see what monomial functions should occur in such a product; things are less clear as regards the coefﬁcients with which they occur. For instance, why is there a 2 in front of m(2,2) in the third formula but not one in front of m(3,2) in the ﬁrst. Things become a good deal better (in my opinion) when one works in the larger ring QSymm ⊃ Symm of quasi-symmetric functions where there is a clear easy to use formula for multiplying quasi-symmetric monomial functions. This is the overlapping shufﬂe product, see 11.26. See also the appendix to this chapter for some more details.

Witt vectors. Part 1

357

9.53. Forgotten symmetric functions. The complete symmetric functions {hλ } form a basis dual to that of the monomial symmetric functions. There also is of course a dual basis to the basis of elementary symmetric functions {eλ }. These are obtained by replacing the hi by the ei in the formulas for the mλ in the hi . There seem to be no ‘nice’ formulas for them, [281], p. 22. 9.54. Power sum symmetric functions. The power sum symmetric functions are deﬁned by ξin (9.55) pn = m(n) = i

Recall that they are the ghost components of the element H(t) = 1 + h1 t + h2 t 2 + h3 t 3 + · · · ∈ (Symm) = 1 + tSymm[[t]] (9.56) The power sum symmetric functions are related to the complete symmetric functions by the Newton relations nhn = pn + h1 pn−1 + h2 pn−2 + · · · + hn−1 p1 which in terms of the generating functions H(t), see (9.56), and P(t) = pi t i (so that p0 is set to zero)

(9.57)

(9.58)

i≥1

can be encoded by the ‘differental’ equation (deﬁning ghost components) P(t) = t

tH (t) d log H(t) = dt H(t)

(9.59)

From (9.59) one readily obtains a formula for the power sum symmetric functions in terms of the complete symmetric functions, i.e. a universal formula for the ghost components of an element 1 + a1 t + a2 t 2 + · · · in terms of its ‘a-coordinates’. pn = (−1)k+1 r1 hr1 hr2 · · · hrk ; ri ∈ N = {1, 2, · · · } (9.60) r1 +r2 +···+rk =n

This formula can readily be inversted by ‘solving’ (9.59) using formal exponentials. For each partition λ (with no zero parts) deﬁne also pλ = pλ1 pλ2 · · · pλn

(9.61)

Then, obviously, given the Newton relations, the pλ form a homogeneous basis over the rationals for the vector space SymmQ . But they are not a basis for Symm itself. For a partition λ = (λ1 , λ2 , · · · , λn ) deﬁne the integer zλ by zλ = 1m1 (λ) m1 (λ)!2m2 (λ) m2 (λ)! · · · nmn (λ) mn (λ)!

358

M. Hazewinkel

where, as before, mi (λ) is the number of parts i in λ. Then ([281], p. 62)

z−1 (1 − ξi ηj )−1 λ pλ (ξ)pλ (η) = λ

(9.62)

i,j

showing that suitably normalized pλ form an orthonormal basis for SymmQ (but not of course for Symm as these normalized pλ are not even elements of Symm). 9.63. On the Witt coordinates. Now consider the transformation from ‘a-coordinates’ of a 1-power-series, i.e. an element of (A), to Witt vector coordinates. This is given by the universal formula, see (9.22)

(1 − xd t d )−1 = 1 + h1 t + h2 t 2 + · · · (9.64) d

Assuredly the xd as deﬁned by (9.64) are symmetric functions and also, obviously, from (9.64) one has xd = hd + (homogenous polynomial of weight d in the h1 , · · · , hd−1 )

(9.65)

and so the xλ = xλ1 xλ2 xλ3 · · ·

(9.66)

form yet another homogenous basis for Symm. It is quite easy to express the hn in terms of the xd . Indeed rewriting the left hand side of (9.61) as

(1 + xi t i + xi2 t 2i + · · · ) (9.67) i

it is immediate that m (λ) m (λ) m (λ) hn = xλ = x1 1 x2 2 x3 3 · · · wt(λ)=n wt(λ)=n

(9.68)

However, ﬁnding a formula the other way seems to be rather messy.31 On the other hand, there is a duality formula. For each symmetric function f(ξ) introduce the shorthand notation f(ξ i ) = f(ξ1i , ξ2i , ξ3i , · · · )

(9.69)32

Now for any partition λ = (1k1 2k2 · · · nkn ) introduce rλ = hk1 (ξ 1 )hk2 (ξ 2 ) · · · hkn (ξ n ) 31 This corresponds to the fact that inverting the Witt vector polynomials appears to be a messy business. 32 This is the (outer) plethysm with perspect to the power sum symmetric function p ; see subsections i

16.76, 18.35 and 18.37 below.

Witt vectors. Part 1

359

It is not difﬁcult to check that these also form a basis of Symm.33 There is now the following duality formula <xλ , rκ > = δλ,κ

(9.70)

i.e. the xλ form a dual or adjoint basis to the one of the rλ . This is due to [350], where there is a quite elegant proof using theorem 9.42. The symmetric functions xn also have an interesting, even remarkable, positivity property: 9.71. Theorem. The symmetric functions −x1 , x2 , · · · , xn , · · · are Schur positive.34 Here ‘Schur positive’ means that when expressed as a Z-linear combination of the Schur symmetric functions all the coefﬁcients are nonnegative. This is stronger than being positive of course. Theorem 9.71 was conjectured by Christoph Reutenauer, [333, 332], and proved by Thomas Scharf and Jean-Yves Thibon, [350], and also by W Doran. Some more investigations relating to the xn are reported in loc. cit. Much remains to be done in my opinion, particularly regarding exploiting the positivity result of theorem 9.71.35 9.72. Integrality lemmata. The miracle of the Witt polynomials (2). For the generalized Witt polynomials wn (X) there is the same integrality theorem as for the p-adic ones. 9.73. Theorem. Let ϕ(X, Y, Z) be a polynomial over the integers in three (or less, or more) commuting indeterminates then there are unique polynomials over the integers ϕn (X1 , X2 , · · · , Xn ; Y1 , Y2 , · · · , Yn ; Z1 , Z2 , · · · , Zn ) = ϕn (X; Y ; Z), n = 1, 2, 3, · · · such that for all n ≥ 1 wn (ϕ1 (X; Y ; Z), · · · , ϕn (X; Y ; Z)) = ϕ(wn (X), wn (Y ), wn (Z))

(9.74)

The proof is (basically) the same as in the p-adic case. The theory of the functor of the big Witt vectors can now be developed starting from this integrality theorem in just the same way as was done for the p-adic Witt vectors in section 5 above. That is deﬁne addition polynomials μS,i (X1 , · · · , Xi ; Y1 , · · · , Yi ) and multiplication polynomials μP,i (X1 , · · · , Xi ; Y1 , · · · , Yi ) by taking the polynomial ϕ(X, Y ) in theorem 9.73 respectively equal to X + Y and XY , so that W W wn (μW S,1 (X; Y ), μS,n (X; Y ), · · · , μS,n (X; Y )) = wn (X) + wn (Y ) W W wn (μW P,1 (X; Y ), μP,2 (X; Y ), · · · , μs,n (X; Y )) = wn (X)wn (Y )

(9.75)

33 This one has a certain amount of special interest in that for any symmetric function f the integrality conditions ∈ Z are equivalent to certain congruences established by Andreas Dress, [118], for testing whether a central function of a symmetric group Sn is a virtual character; see Thomas Scharf and Jean-Yves Thibon, [350]. 34 There is no printing error here; there really is just one minus sign. 35 Quite generally the systematic use of (partial) orderings and positive properties in Hopf algebra theory (and hence Witt vector theory) has started only fairly recently.

360

M. Hazewinkel

and deﬁne an addition and multiplication on Witt vectors a = (a1 , a2 , a3 , · · · ) by W W (a1 , a2 , a3 , · · · ) +W (b1 , b2 , b3 , · · · ) = μW s,1 (a; b), μs,2 (a; b), μs,3 (a; b), · · ·

W W (a1 , a2 , a3 , · · · ) ·W (b1 , b2 , b3 , · · · ) = μW P,1 (a; b), μP,2 (a; b), μP,3 (a; b), · · · (9.76)

This then deﬁnes the functorial ring W(A) with multiplication and addition given by (9.76) with zero element (0, 0, 0, · · · ) and unit element (1, 0, 0, 0, · · · ). 9.77. Theorem 9.73 is only one of a slew of integrality theorems. One of my favourites is the ‘functional equation lemma’ which is of great use in formal group theory; particularly in the construction of universal formal groups, see [192], Chapter 1, §2.3. 9.78. Ingredients for the functional equation integrality lemma. The ingredients for the functional equation lemma are the following A ⊂ K, σ: K −→ K, p ⊂ A, p, q, s1 , s2 , s3 , · · ·

(9.79)

Here A is a subring of a ring K, σ is a ring endomorphism of K, a is an ideal in A, p is a prime number and q is a power of p, and the si are elements of K. These ingredients are supposed to satisfy the following conditions σ(A) ⊂ A, σ(a) ≡ aq mod p for all a ∈ A, p ∈ p, si b ∈ A for all b ∈ p (9.80) and also pr b ⊂ p ⇒ pr σ(b) ⊂ p for all r ∈ N, b ∈ K

(9.81)

a property that is automatically satisﬁed if the ideal p = (c) is principal and σ(c) = uc for some unit of A. Here are three examples of such situations. There are many more. A = Z, K = Q, σ = id, q = p, p = pZ, si ∈ p−1 Z

(9.82)

A = Z(p) , K = Q, σ = id, q = p, p = pZ(p) , si ∈ p−1 Z(p)

(9.83) p

p

A = Z[V1 , V2 , · · · ], K = Q[V1 , V2 , · · · ], σf(V1 , V2 , · · · ) = f(V1 , V2 , · · · ), q = p, p = pA, si ∈ p−1 A

(9.84)

9.82. Constructions for the functional equation lemma. Now, let g(X) = b1 X + b2 X2 + · · · be a power series with coefﬁcients in A. Using the ingredients (9.78) construct from it a new power series by the recursion formula (or functional equation) fg (X) = g(X) +

∞ i=1

i

si σ∗i fg (Xq )

(9.85)

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where σ∗i fg (X) is the power series obtained from fg (X) by applying σ i to the coefﬁcients of fg (X). Three examples of power series obtained in this way are: X + p−1 Xp + p−2 Xp + p−3 Xp + · · · 2

log(1 + X) =

∞

3

(9.86)

n−1 (−1)n+1 Xn

(9.87)

n=1 p

X + p−1 V1 Xp + (p−2 V1 V1 + p−1 V2 )Xp p

p2

2

p

p2

+ (p−3 V1 V1 V1 + p−2 V1 V2 + p−2 V2 V1 + p−1 V3 )Xp + · · · 3

(9.88)

9.89. Functional equation integrality. Lemma. Let A ⊂ K, σ, p ⊂ A, p, q, s1 , s2 , s3 , · · · be as in (9.77) and be such that (9.80) and (9.81) hold. Let g(X) = b1 X + b2 X2 + · · · and g(X) = b1 X + b2 X2 + · · · be two power series with coefﬁcients in A and let b1 be invertible. Let this time f −1 (X) denote the functional inverse of f(X) (for a power series with zero constant term and nonzero coefﬁcient in degree 1, deﬁned by f −1 (f(X)) = X. Then The power series Fg (X, Y ) = fg−1 (fg (X) + fg (Y )) has its coefﬁcients in A (9.90) The power seriesfg−1 (fg (X)) has its coefﬁcients in A

(9.91)

If h(X) is a power series with zero constant term over K, then h(X) ≡ g(X) mod pr A[[X]] ⇔ fg (h(X)) ≡ fg (g(X)) mod pr A[[X]] (9.92) Statement (9.90) says that Fg (X, Y ) is a (one dimensional) formal group over A with logarithm fg (X). Example (9.88) is the logarithm of the universal one dimensional p-typical formal group. There are also more dimensional and inﬁnite dimensional versions and these can be used to construct universal formal groups and also the Witt vectors. For all these statements and the appropriate deﬁnitions see [192]. Another application of the functional equation lemma is the following statement, especially useful in the Witt vector context. 9.93. Lemma. [192], lemma 17.6.1, page 137; ghost component integrality lemma. Let A be a characteristic zero ring with endomorphisms ϕp for all prime numbers p such that ϕp (a) ≡ ap for all a ∈ A. Then for a given sequence b1 , b2 , · · · in A there exists another sequence x1 , x2 , · · · in A such that wn (x) = bn for all n (i.e. the sequence b1 , b2 , · · · lies in the image of the ghost mapping) if and only if ϕp (bn ) ≡ bnp mod(pvp (n)+1 )

(9.94)

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M. Hazewinkel

Here, as before, see example 6.10, vp denotes the p-adic valuation on Z. A vector with components in A which is in the image of w: W(A) −→ AN, (a1 , a2 , · · · ) → (w1 (a), w2 (a), · · · ) will be called a ghost-Witt vector over A. As far as I know, this lemma, in this form, ﬁrst appeared in [86]. Pierre Cartier attributes it to Bernard Dwork and Jean Dieudonné, [107], proposition 1; [132], lemma 1. The formulations there are a bit different. In e.g. the case A = Z where one takes all the endomorphisms ϕp to be the identity, the integrality condition (9.94) can be given the following quite elegant formulation. Write down the Mobius inversion transform of the sequence b1 , b2 , · · · , i.e. the sequence cn = bn/d μ(d) (9.95) d|n

Then the sequence b1 , b2 , · · · satisﬁes (9.94) if and only if cn ≡ 0 mod n for all n. This remark seems to be due to Albrecht Dold, [113]. Here the Möbius function μ: N −→ Z is deﬁned by μ(1) = 1, μ(n) = (−1)r if n is the product of precisely r different primes and μ(n) = 0 otherwise. Still another formulation of the integrality condition is n b(i,n) ≡ 0 mod n (9.96) i=1

where (i, n) denotes the greatest common divisor of i and n. The (integrality aspects of the) theory of Witt vectors can be developed solely on the basis of this lemma 9.93. This is how things are done in [212]. 9.97. Witt vectors over the integers and ﬁxed points and ﬁxed point indices of iterated mappings. In [113], Dold proves that a sequence of integers s = (s1 , s2 , s3 , · · · ) is the sequence of ﬁxed point indices of the iterates of a continuous mapping if and only if the formal power series ∞ si i exp − t i i=1

has integral coefﬁcients; that is iff s is the ghost vector of a Witt vector over the integers. This is done using the integrality criterium (9.96). More concretely, call a sequence of nonnegative integers exactly realizable if there is a set X and a map f : X −→ X such that the number of ﬁxed points of the n-th iterate f n of f is exactly an . Then a sequence is exactly realizable if and only if the numbers μ(d)an/d d|n

are nonnegative and divisible by n, [330]. Similar things happen when studying iterates of mappings of the unit interval into itself, i.e. in the part of dynamical system and chaos theory that deals with such mappings, see [70, 292].

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The ﬁrst extensive treatment that I know of the Witt vector constructions based basically totally on the symmetric function point of view is by Pierre Cartier, [86]. 10. The Hopf algebra Symm as the representing algebra for the big Witt vectors Recall that Symm = Z[h1 , h2 , h3 , · · · ] = Z[h] represents the functor and hence the functor W of the big Witt vectors (A) = {a(t) = 1 + a1 t + a2 t 2 + a3 t 3 + · · · : ai ∈ A} = CRing(Z[h], A) (10.1) The fact that this is an Abelian group valued functor means that there is a comultiplication making Symm a coalgebra object in the category CRing of commutative unital rings, which in turn means that there is a comultiplication, and a co-unit μs: Symm −→ Symm ⊗ Symm,

εs: Symm −→ Z,

(10.2)

that are ring morphisms and that make (Symm, μs , εs )36 a co-associative, cocommutative co-unital coalgebra. See [293] for a lot of material on coalgebras. Because the addition on the set (A) is deﬁned as multiplication of power series and the unit element is the power series 1, the morphisms μs , εs are given by μs (hn ) = 1 ⊗ hn +

n−1

hi ⊗ hn−i + hn ⊗ 1, εs (hn ) = 0,

n≥1

(10.3)

i=1

(It is often convenient to deﬁne h0 = 1 so that the multiplication can be written i+j=n hi ⊗ hj .) Writing m: Symm ⊗ Symm −→ Symm, e : Z −→ Symm

(10.4)

for the multiplication and unit element of Symm, in total there is a bialgebra structure (Symm, m, e, μs , εs ). Finally there is an antipode ιs : Symm −→ Symm

(10.5)

The existence of an antipode comes for free in the present case, see below. The antipode is an anti ring morphism (and hence in the present case a ring morphism because of commutativity). The antipode is determined by ιs (hn ) = −hn

(10.6)

All this makes the total structure (Symm, m, e, μs , εs , ιs ) a Hopf algebra over the integers. For the basic theory of Hopf algebras, see [95]. 36 The index ‘S’ here refers to the matter that these are the ring morphisms that give the sum part of the Witt vector functor.

364

M. Hazewinkel

10.7. Skew Schur functions. (Very partial symmetric function formularium (2)). Let κ, λ be two partitions and let sκ , sλ be the corresponding Schur funcv tions. As the Schur functions form an orthogonal basis, there are coefﬁcients cκ,λ such that v v cκ,λ sv or, equivalently, sκ sλ , sv = cκ,λ (10.8) sκ s λ = v

v have a combinatorial interpretation, see [281], The multiplicity coefﬁcients37 cκ,λ Chapter 1, §9, p. 143, and are, hence, nonnegative integers. Deﬁne the skew Schur function sκ/λ by κ sκ/λ , sv = sκ , sλ sv , or, equivalently, sκ/λ = cλ,v sv (10.9) v

It turns out that sκ/λ = 0 unless λ ⊂ κ which by deﬁnition means that λi ≤ κi for all i.38 There is a determinantal formula for the skew Schur functions as follows sκ/λ = det(hκi −λj −i+j )i,j

(10.10)

In terms of skew Schur functions the comultiplication of Symm can be written μS (sκ ) = sκ/λ ⊗ sλ (10.11) λ

as follows from the duality formula xy, z = x ⊗ y, μs (z) in Symm. 10.12. Grading. Symm is a graded Hopf algebra. That is, as an Abelian group it is a direct sum ∞

Symm = ⊕ Symmn n=0

(10.13)

where the homogenous part of degree (or weight) n is spanned by the monomials in the hi of weight n where hi has weight i; i.e. by the monomials hλ with wt(λ) = n, and that moreover the multiplication and comultiplication and unit and co-unit are compatible with the grading, which means m(Symmi ⊗ Symmj ) ⊂ Symmi+j , e(Z) ⊂ Symm0 μs (Symmn ) ⊂ ⊕ Symmi ⊗ Symmj , ε(Symmn ) = 0 i+j=n

for n ≥ 1

ιs (Symmn ) ⊂ Symmn Note that Symm0 = Z and from the point of view of this component e and ε identify with the identity on this component. 37 In the representation theoretic incarnation of Symm, see section 18 below, these coefﬁcients turn up as multiplicities of irreducible representations (Littlewood–Richardson rule). 38 κ ⊃ λ is the case if and only if the diagram of κ contains the diagram of λ; whence the notation.

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A graded Hopf algebra is connected if its degree 0 component is of rank 1. It is easy to see that for a connected graded bialgebra there is a unique antipode making it a Hopf algebra. 10.14. Antipodes on connected graded bialgebras. Indeed, as H0 = Z (or whatever ring once is working over), an antipode must be the identity on H0 . Further, if x is primitive, i.e. μH (x) = 1 ⊗ x + x ⊗ 1, the deﬁning requirements for an antipode say that ι(x) = −x. Further, if i is the smallest integer ≥ 1 such that Hi = 0 all elements of Hi must be primitive by degree considerations (and the counit properties). Finally if the antipode is known on all the Hj with j < n and x ∈ Hn , id ⊗ ι and ι ⊗ id are known on all the terms of μH (x) except 1 ⊗ x and x ⊗ 1. The deﬁning requirements for an antipode then immediately give a formula for ι(x) (two formulas really). They must give the same result because anitpodes (if they exist) are unique (by the same argument that is used to show that inverses in groups are unique). 10.15. Primitives of Symm. For every Hopf algebra it is important to know its primitives. That is the elements x that satisfy μ(x) = 1 ⊗ x + x ⊗ 1.

(10.16)

These form a module over the ring over which the Hopf algebra is deﬁned (in this case Z) and have a Lie algebra structure under the commutator product [x, y] = xy − yx. Because of commutativity this Lie product is zero in the present case. By deﬁnition, p1 = h1 is a primitive. With induction, from the Newton relations (9.57) it follows immediately that all the pn are primitives. It is not difﬁcult to see that these form a basis over Z for the module of primitives.39 So the comultiplication and counit of Symm satisfy μs (pn ) = 1 ⊗ pn + pn ⊗ 1, εs (pn ) = 0

(10.17)

and as the power sum symmetric functions form a free polynomial basis (over Q) for the symmetric functions this is a perfectly good description of the Hopf algebra Symm (once it is known that it is in fact deﬁned over the integers). Formulas (10.17) of course amount to completely the same thing as saying that at the ghost component level addition is coordinate-wise. 10.18. Liulevicius theorem, [269, 268]. There are precisely four graded Hopf algebra automorphisms of Symm. They are functorially given by identity, a(t) → a(−t), a(t) → a(t)−1 , a(t) → a(−t)−1

(10.19)

and they form the Klein 4-group. 39 First observe that if an element is primitive its different homogeneous summands must all be primitives. next observe that a term in the coproduct of a monomial of length n can only cancel against one from another term of the same length; continue by using lexicographic order on exponent sequences. Finally, as pn ≡ (−1)n+1 hn1 mod (h2 , h3 , · · · ) (also from the Newton relations with induction), the pn are not divisible by any natural number other than 1 and as the different pn are of different degree no sum of them is divisible by any natural number other than 1.

366

M. Hazewinkel

The proof is quite simple and it is surprising that this theorem was discovered so late.40 Over the rationals of course the situation is quite different and the group of automorphisms is quite large viz a countable sum of copies of the multiplicative group of the rationals. 10.20. Here is the proof. The power sum symmetric functions are primitives for the Hopf algebra structure meaning that μs (pi ) = 1 ⊗ pi + pi ⊗ 1

(10.21)

Moreover every homogeneous primitive of weight i is a scalar multiple of pi . Let ϕ be a graded automorphism. Automorphisms of Hopf algebras must take primitives into primitives and as a graded automorphism preserves the grading it must be the case that ϕ(pi ) = api with a ∈ {1, −1} because ϕ is an invertible morphism. The last three automorphisms named in (10.19) take on p1 , p2 respectively the values −p1 , p2 ; −p1 , −p2 ; p1 , −p2 and so composing with a suitable one from the four automorphisms (10.19) it remains to analyze the case that ϕ(p1 ) = p1 , ϕ(p2 ) = p2 Suppose with induction that it has been proved that ϕ(pi ) = pi , i < n ≥ 3. Look at the Newton relations (see (9.57)) nhn = pn + h1 pn−1 + · · · + hn−1 p1 ,

(10.22)

Because the hi are polynomials in the p1 , p2 , · · · , pi (albeit with rational coefﬁcients), ϕ(hi ) = hi , i < n. So applying ϕ to (10.16) and subtracting the result from (10.16) one ﬁnds n(hn − ϕ(hn )) = pn ± pn which because n ≥ 3 and ϕ(hn ) must be an integral polynomial of weight n, and pn is not divisible by any integer > 1, is only possible if on the right hand side of (10.13) the minus sign applies; i.e. only if ϕ(pn ) = pn . 10.23. Product comultiplication. Besides the Abelian group structure on (A) and W(A) there is also the product of Witt vectors and power series. These are also functorial and (hence) given by universal polynomials, i.e. by algebra morphisms μp: Symm −→ Symm ⊗ Symm, εp : Symm −→ Z 40 This has most probably to do with the unwarranted and regrettable tendency of mathematicians to

think primarily of Hopf algebras, etc. as things over a ﬁeld of characteristic zero.

Witt vectors. Part 1

367

and these give a second bialgebra structure (Symm, m, e, μp , εp ) on Symm. Because the functorial unit on the rings (A) is given by the power series 1 + t + t 2 + t 3 + · · · the co-unit morphism of rings εp is given by εp (hn ) = 1,

n = 1, 2, 3, · · ·

(10.24)

The formula for the morphism governing the multiplication is less easy to describe; certainly explicitly. Various descriptions will be given later in the present section 10. There is no antipode for the second comultiplication of course (otherwise Symm would deﬁne a ﬁeld valued functor). Putting all these structure morphisms together there results the object (Symm, m, e, μs , εs , ιs , μp , εp ) which is a coring object in the category CRing of commutative unital rings. One axiom that such an object much satisfy is distributivity on both the left and the right of the second comultiplication over the ﬁrst in the category CRing (where the appropriate codiagonal map is the multiplication on a ring). For distributivity on the left this means that the following diagram must be commutative

ms

Symm

Symm

Symm

ms Symm

Symm

Symm

Symm τ

Symm id

Symm

Symm

¬

id

Symm

¬

¬

Symm

¬

Symm

Symm

mp Ä mp

mp

id

m id id

Symm

Symm

where, as usual τ : a ⊗ b → b ⊗ a is the twist morphism. There is a similar diagram for distributivity on the right. 10.25. Various descriptions of the second comultiplication morphism on Symm. At ﬁrst sight the easiest description is of course in terms of the power sum symmetric functions as μP (pn ) = pn ⊗ pn

(10.20)

but for a variety of reasons this one is of very limited use. It is of course the deﬁnition of the multiplication in terms of ghost components. are the three descriptions that follow from the three expansions of Much better (1 − ξi ηj )−1 given in (9.62), (9.44), (9.45). These give the second comultiplication formulas hλ ⊗ mλ (10.21) μP (hn ) = λ

μP (hn ) =

λ

μP (hn ) =

λ

z−1 pλ ⊗ p λ λ

(10.22)

sλ ⊗ s λ

(10.23)

368

M. Hazewinkel

where in all three formulas the sum is over all partitions of n. Explicitly the ﬁrst few multiplication polynomials are μP (h1 ) = h1 ⊗ h1 , μP (h2 ) = 2h2 ⊗ h2 − (h2 ⊗ h1 2 + h1 2 ⊗ h2 ) + h1 2 ⊗ h1 2 μP (h3 ) = 3h3 ⊗ h3 − 3(h3 ⊗ h2 h1 + h2 h1 ⊗ h3 ) + (h3 ⊗ h1 3 + h1 3 ⊗ h3 ) − (h2 h1 ⊗ h1 3 + h1 3 ⊗ h2 h1 ) + h1 3 ⊗ h1 3

(10.24)

10.25. Still other descriptions of the second comultiplication morphism and the (functorial) multiplication on W(A) will be given below, in sections 15 and 11 respectively. 10.26. Product comultiplication vs inner product. Theorem 9.42, in particular the expansion formula (9.44), compared with the deﬁnition of the multiplication on the Witt vectors determined by (1 − ξt)−1 ∗ (1 − ηt)−1 = (1 − ξηt)−1 suggest that there are nontrivial interrelations between the inner product on Symm and the second comultiplication (= product comultiplication) on Symm. And so there are. For instance cocommutativity of the product comultiplication and symmetry of the inner product are equivalent. At this stage and in this setting this is just an amusing remark. There may be more to it in other contexts. The inner product is deﬁned by formula (9.41) hλ , mμ = δλ,μ Write the monomial symmetric functions as (integer) linear combinations of the hλ : λ mμ = aμ hλ λ

so that μ mμ , mμ = aμ

Now (by (9.44)) μP (hn ) =

(10.27)

hλ ⊗ mλ =

aλλ hλ ⊗ hλ

(10.28)

wt(λ)=n

By cocommutativity of the product comultiplication, also

mλ ⊗ hλ = aλλ hλ ⊗ hλ μP (hn ) = wt(λ)=n

Comparison ot the two expressions (10.28) and (10.29) gives, see (10.27),

aλλ = aλλ

i.e. symmetry of the inner product. The argument also runs the other way.

(10.29)

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369

11. QSymm, the Hopf algebras of quasisymmetric functions and NSymm, the Hopf algebra of noncommutative symmetric functions When looking at various universality properties of the Witt vectors and Symm (which is the topic of the next section) one rapidly stumbles over a (maximally) non commutative version, NSymm, and a (maximally) non cocommutative version, QSymm. This section is devoted to a brief discussion of these two objects. Somehow a good many things become easier to see and to formulate in these contexts (including certain explicit calculations). As I have said before, e.g. in [200], p. 56; [199], Ch 1, p. 1, once one has found the right non commutative version, things frequently become more transparent, easier to understand, and much more elegant. 11.1. The Hopf algebra of non commutative symmetric functions. Let Z = {Z1 , Z2 , Z3 , · · · } be a countably inﬁnite set of (noncommuting) indeterminates. As an algebra the Hopf algebra of noncommutative symmetric functions is simply the free associative algebra in these indetermines over the integers NSymm = ZZ1 , Z2 , Z3 , · · · = ZZ

(11.2)

The coalgebra structure is given on the generators Zn by μ(Zn ) = Zi ⊗ Zj = 1 ⊗ Zn + Z1 ⊗ Zn−1 + Z2 ⊗ Zn−2 i+j=n

+ · · · + Zn−1 ⊗ Z1 + Zn ⊗ 1

(11.3)

where the notation Z0 = 1 is used in the expression in the middle. For every word over the natural numbers α = [a1 , a2 , · · · , am ], ai ∈ N = {1, 2, 3, · · ·} Zα = Za1 Za2 · · · Zam

(11.4)

denotes the corresponding (non commutative) monomial. This incudes Z[] = 1. These form a basis of the free Abelian group underlying NSymm. The co-unit morphism is given by ε(Zn ) = 0, n = 1, 2, 3, · · · , ε Z[ ] = 1 (11.5) Give Zn degree (or weight) n, so that the weight (or degree) of a monomial Zα is a1 + a2 + · · · + am = wt(α). Then, obviously, NSymm is a graded Abelian group NSymm = ⊕ NSymmn n≥0

(11.6)

with the homogeneous part of degree n spanned by the monomials (11.4) of weight n. Then, obviously, (Symm, m, e, μ, ε) is a graded, connected (meaning that NSymm0 = Z) bialgebra so that there is (for free, see 10.14) a suitable antipode making NSymm a connected, graded, non commutative, cocommutative Hopf algebra. There is a natural surjective morphism of graded Hopf algebras NSymm −→ Symm, Zn → hn

(11.7)

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M. Hazewinkel

that exhibits Symm as the maximal commutative quotient Hopf algebra of NSymm. The kernel of (11.7) is the commutator ideal generated by all elements of the form Zi Zj − Zj Zi , i, j ∈ N. The ﬁrst paper to study NSymm in depth was probably [162]. It was immediately followed by a slew of other publications [54, 129, 130, 128, 216, 214, 219, 249, 251, 250, 386, 384, 200, 195, 198, 193, 201]. 41 11.8. Divided power sequences. Curves. Given a Hopf algebra H a divided power sequence, or curve, in H is by deﬁnition a sequence of elements γ = (d0 = 1, d1 , d2 , · · ·) such that μH (dn ) =

di ⊗ d j

(11.9) (11.10)

i+j=n

If H is the covariant bialgebra of a formal group F this translates into a mapping of the formal afﬁne line into the formal scheme (variety) underlying F , whence the terminology curve.42 A curve is often denoted by its generating power series γ (t) = 1 + d1 t + d2 t 2 + d3 t 3 + · · ·

(11.11)

Two such power series can be multiplied and the result is again a curve, i.e a power series in t of which the coefﬁcients satsify (11.10). This deﬁnes a functor Curve: Hopf −→ Group

(11.12)

from the category of Hopf algebras to the category of groups. An example of a curve is the universal curve Z(t) = 1 + Z1 t + Z2 t 2 + · · · ∈ Curve(NSymm)

(11.13)

and it is immediate from the deﬁning property (11.10) and the freeness of NSymm as an associative algebra that 11.14. Theorem (universality property of NSymm). For every curve γ(t) in a Hopf algebra H there is a unique morphism of Hopf algebras NSymm −→ H that takes the universal curve Z(t) to the given curve γ(t). That is, the functor Curve is represented by NSymm.43 The group operation on curves corresponds to the convolution product on Hopf(NSymm, H). When the Hopf algebra H is commutative this theorem translates into the following. 41 Many of these papers are about noncommutative symmetric functions over a ﬁeld of characteristic zero; here, of course, it is NSymm over the integers that is really important (here and elsewhere). 42 There is also another notion in algebra that goes by the name ‘divided powers’ viz the presence of a sequence of operators (usually on an ideal in a commutative ring) that behave like sending an element x to (n!)−1 xn . Hence the term curve, rather than divided power sequence (DPS) is preferred for the notion deﬁned by (11.9), (11.10). 43 See also [149].

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371

11.15. Theorem. (Second universality property of the Witt vectors). Let γ(t) be a curve in a commutative Hopf algebra H. Then there is a unique morphism of Hopf algebras Symm −→ H that takes the curve h(t) = 1 + h1 t + h2 t 2 + · · · ∈ Curve(Symm)

(11.16)

into the given curve γ(t). In the context of commutative formal groups with H the covariant bialgebra of a commutative formal group this is known as Cartier’s ﬁrst theorem. 11.17. The Hopf algebra QSymm of quasi-symmetric functions. The next sections deal with another generalization of Symm, dual to NSymm and containing Symm. 11.18. Compositions and partitions. A composition is a word (of ﬁnite length) over the natural numbers. Here a composition will be written α = [a1 , a2 , · · · , am ], ai ∈ N = {1, 2, 3, · · · }

(11.19)

A composition (11.19) is said to have length m and weight wt(α) = a1 +a2 +· · ·+am . Associated to a composition α is a partition λ which consists of the aj re-arranged in nonincreasing order. Or, equivalently, λ = 1m1 (α) 2m2 (α) 3m3 (α) · · · where mj (α) is the number of entries of α equal to j. 11.19. Monomial quasi-symmetric functions. Consider again the ring of polynomials, i.e. power series of bounded degree44 in a countable inﬁnity of commuting indeterminates, i.e. an element of Z [ξ1 , ξ2 , ξ3 , · · ·]. Such a polynomial cα,ι1 ,··· ,im ξia11 ξia22 · · · ξiamm (11.20) where the sum runs over runs over all compositions α = [a1 , · · · , am ] and all index sequences i1 < · · · < im of the same length as α, is said to be quasi-symmetric if cα, ι1 ,··· , im = cα,j1 ,··· ,j m for all index sequences i1 < · · · < im and j1 < · · · < jm (11.21) For a given composition α = [a1, · · · , am ] the associated monomial quasi-symmetric function is ξia11 ξia22 · · · ξiamm (11.22) [a1, · · · , am ] = i1
It is denoted by the same symbol. For example in four variables [1, 2] = ξ1 ξ22 + ξ1 ξ32 + ξ1 ξ42 + ξ2 ξ32 + ξ2 ξ42 + ξ3 ξ42 [1, 2, 1] = ξ1 ξ22 ξ3 + ξ1 ξ22 ξ4 + ξ1 ξ32 ξ4 + ξ2 ξ32 ξ4 [3, 2, 1] = ξ13 ξ22 ξ3 + ξ13 ξ22 ξ4 + ξ13 ξ32 ξ4 + ξ23 ξ32 ξ4 44 See the appendix.

(11.23)

372

M. Hazewinkel

Note that the monomial symmetric function mλ deﬁned by a partition λ is the sum of all distinct monomial quasi-symmetric functions α whose associated partition is λ. For example m(2,1) = [2, 1] + [1, 2], m(2,1,1) = [2, 1, 1] + [1, 2, 1] + [1, 1, 2] m(3,2,1) = [3, 2, 1] + [3, 1, 2] + [2, 3, 1] + [2, 1, 3] + [1, 3, 2] + [1, 2, 3]

(11.24)

The monomial quasi-symmetric functions form a free basis of the free Abelian group QSymm of all quasi-symmetric functions. This group is graded by assigning to the monomial quasi-symmetric function α = [a1 , · · · , am ] the weight wt(α). Sum and product of quasi-symmetric functions are again quasi-symmetric, so that QSymm is a ring. The ring Symm is a subring with the inbedding hn = α (11.25) wt(α)=n

11.26. Overlapping shufﬂe product. It is useful to have an explicit direct recipe in terms of compositions for the multiplication of monomial quasi-symmetric functions. This is given by the overlapping shufﬂe product. Let α = [a1 , a2 , · · · , am ] and β = [b1 , b2 , · · · , bn ] be two compositions or words. Take a ‘sofar empty’ word with n + m − r slots where r is an integer between 0 and min{m, n}, 0 ≤ r ≤ min{m, n}. Choose m of the available n + m − r slots and place in it the natural numbers from α in their original order; choose r of the now ﬁlled places; together with the remaining n + m − r − m = n − r empty places these form n slots; in these place the entries form β in their original order; ﬁnally, for those slots which have two entries, add them. The product of the two words α and β is the sum (with multiplicities) of all words that can be so obtained. So, for instance [a, b][c, d] = [a, b, c, d] + [a, c, b, d] + [a, c, d, b] + [c, a, b, d] + [c, a, d, b] + [c, d, a, b] + [a + c, b, d] + [a + c, d, b] + [c, a + d, b] + [a, b + c, d] + [a, c, b + d] + [c, a, b + d] + [a + c, b + d] (11.27) [1][1][1] = 6[1, 1, 1] + 3[1, 2] + 3[2, 1] + [3]

(11.28)

It is easy to see that the recipe given above gives precisely the multiplication of (the corresponding basis) quasi-symmetric functions. The shufﬂes with no overlap of a1 , · · · , am ; b1 , · · · , bn correspond to the products of the monomials that have no ξj in common; the other terms arise when one or more of the ξ‘s in the monomials making up α and β do coincide. In example (11.27) the ﬁrst six terms are the shufﬂes; the other terms are ‘overlapping shufﬂes’. The term shufﬂe comes from the familiar riﬂe shufﬂe of cardplaying; an overlapping shufﬂe occurs when one or more cards from each deck don’t slide along each other but stick edgewise together; then their values are added.

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11.29. Hopf algebra structure on QSymm. Now introduce on QSymm the comultiplication ‘cut’: μQSymm ([a1 , a2 , · · · , am ]) = 1 ⊗ [a1 , a2 , · · · , am ] +

n−1 [a1 , · · · , ai ] ⊗ [ai+1 , · · · , am ]

(11.30)

i=1

+[a1 , a2 , · · · , am ] ⊗ 1 and the counit

εQSymm (α) =

1

if lg(α) = 0

0

if lg(α) ≥ 1

(11.31)

Here recall that the only composition of length zero is the empty composition 1 = [ ] which is the unit for the overlapping shufﬂe multiplication. Write mQSymm for the overlapping shufﬂe multiplication and eQSymm for the unit morphism determined by 1 → [ ] = 1. Then 11.32. Theorem. The structure (QSymm, mQSymm , eQSymm , μQSymm , εQSymm ) is a commutative non cocommutative graded connected bialgebra and hence deﬁnes a Hopf algebra. The antipode again comes for free once the the bialgebra statement has been proved because QSymm is connected graded. The only thing in proving the bialgebra statement which is not dead easy is the veriﬁcation that the comultiplication is an algebra morphism for the overlapping shufﬂe product. Even that can be avoided by using graded duality as described below. 11.33. Duality between. NSymm and QSymm. Introduce the nondegenerate graded pairing , : NSymm × QSymm −→ Z, Zα , β = δα,β

(11.34)

With respect to this pairing the multiplication (resp. comultiplication) on NSymm is dual to the comultiplication (resp. multiplication) on QSymm. Similarly the counit (resp. unit) on NSymm is dual to the unit (resp. counit) on QSymm. These statements amount to the following four formulas. m NSymm (Zα ⊗ Zβ ), γ = Zα ⊗ Zβ , μ QSymm (γ) μ NSymm (Zα ), β ⊗ γ = Zα , m QSymm (β ⊗ γ) e NSymm (1), β = 1, ε QSymm (β)

(11.35)

ε NSymm (α), 1 = α, e QSymm (1) Where , on Z is the innerproduct r, s = rs. Formulas (11.35) easily imply that if NSymm is a bialgebra, so is QSymm (and vice-versa). The antipodes on the two

374

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Hopf algebras are (as a consequence) also dual to each other: ιNSymm (Zα ), β = Zα , ιQSymm (β) 11.36. Primitives of QSymm. It is an easy exercise to show that the primitives for the ﬁrst comultiplication, the ‘sum comultiplication’, i.e. the elements for which μS (x) = 1 ⊗ x + x ⊗ 1 are precisely the linear combinations of the words of length 1.45 11.37. Autoduality of Symm. Under (graded) duality a sub Hopf algebra of a Hopf algebra H corresponds to a quotient Hopf algebra of the graded dual. To avoid notational confusion write temporarily Symm’ for the algebra of symmetric functions as a subalgebra of QSymm and retain the notation Symm = Z[h] for the algebra of symmetric functions as a quotient algebra of NSymm. The Hopf ideal J in NSymm such that Nsymm/J = Symm is the commutator ideal spanned by all elements of the form Zα (Zi Zj − Zj Zi )Zβ . It is easy to check that a quasisymmetric function has pairing 0 (under (11.34) with all these elements if and only if it is symmetric. Thus the sub Hopf algebra of QSymm corresponding to the quotient Symm of NSymm is Symm’. Also the induced pairing is hλ , mκ = δλ,κ

(11.38)

which is the inner product on Symm as deﬁned in 9.40.46 Moreover as algebras Symm and Symm’ are isomorphic under the isomorphism hn → en . To prove that Symm and Symm’ are isomorphic as Hopf algebras, it remainsto check that this isomorphism takes the given comultiplication on Symm, hn → i+j=n hi ⊗ hj into the comultiplication induced on Symm’ by ‘cut’ the comultiplication on QSymm. But en = [1, 1, 1, · · · , 1] as an element of Symm’ ⊂ QSymm and so ‘cut = μQSymm’ on en takes the value ei ⊗ e j μQSymm (en ) = i+j=n

and things ﬁt perfectly. 11.39. The second comultiplication on QSymm. There is a second comultiplication (morphism) on QSymm that is distributive over the ﬁrst one on the right but not on the left. Here is the deﬁnition. Let α = [a1 , a2 , · · · , am ] be a composition. A (0, α)matrix is a matrix whose entries are either zero or one of the ai , which has no zero columns or zero rows, and in which the entries a1 , a2 , · · · , am occur in their original 45 The situation for the primitives of NSymm is very different. Over the rationals the primitives form the free Lie algebra in countably many generators. Over the integers the description is still more involved and was only recently found; see [198]. 46 This is one main reason for working with the complete symmetric functions rather than with the elementary ones.

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order if one orders the entries of a matrix by ﬁrst going left to right through the ﬁrst row, then left to right through the second row, etc. For a matrix M let column(M) be the vector of column sums and row(M) the vector of row sums. For instance ⎛ ⎞ 0 1 0 3 ⎜ ⎟ M = ⎝1 2 0 1⎠ (11.40) 0 0 1 0 is a (0, [1, 3, 1, 2, 1, 1])-matrix with column(M) = [1, 3, 1, 4] and row(M) = [4, 4, 1]. The second comultiplication on QSymm is now given by μP (α) = row(M) ⊗ column(M) (11.41) M is a (0,α)-matrix

Restricted to Symm ⊂ QSymm, this describes the second comultiplication on Symm in combinatorial terms. To specify it, it sufﬁces to do this on the elementary symmetric polynomials (because it is a ring morphism). i.e. to use the recipe for compositions of the form α = [1, 1, 1, · · · , 1]. For these compositions, the matrices involved are what are usually called (0, 1)-matrices. Probably the most famous theorem on (0, 1)-matrices is the Gale-Ryser theorem. Thus it would appear likely that there is some connection between Witt vectors (via symmetric functions) and the Gale Ryser theorem, [150, 345]. And indeed there is (via the additional link of the symmetric functions with the representations of the symmetric groups). It is the Snapper Liebler-Vitale Lam theorem, [196, 254]. This will be described in a little more detail in subsection 18.33 below. 11.42. Distributivity properties of μP over μS . The second comultiplication morphism μP of (11.39) is distributive on the right over the ﬁrst comultiplication morphism μS but not on the left. Quite generally let (H, m, e, μ, ε) be a bialgebra equipped with a second comultiplication morphism of rings μP :H −→ H ⊗ H. For this second comultiplication to be distributive on the right over the ﬁrst one (in the category of rings) one needs the following diagram (11.43) to commute. (This is precisely what is needed to ensure that the functor represented by H, A → CRing(H, A) have as values groups (denoted additively even when they are noncommutative) (as in the case at hand)) equipped with an additional multiplication such that (a + b)c = ab + bc.)

H

¬

HÄHÄHÄH

HÄHÄH

id Ä tw Ä id

¬

HÄH Ä id

HÄH p Ä p ¬

¬

p

id Ä id Ä m

HÄHÄHÄH

(11.43)

376

M. Hazewinkel

Here is a proof that this diagram is commutative in the case at hand. Take a composition α and let M be a (0, α)-matrix. Applying μ⊗ id to row(M) ⊗ column(M) is the same thing as cutting the matrix M horizontally to obtain two blocks M1 , M2 with the same number of columns stacked on top of each other. All terms in the lower left hand corner of diagram (11.43) coming from α thus are of the form row(M1 ) ⊗ row(M2 ) ⊗ column(M). Now in μ(α) (top horizontal arrow in the diagram) consider the term α1 ⊗ α2 for which the length of α1 is the same as the number of rows of M1 . Let M1 , M2 be the matrices obtaine from M1 , M2 by removing zero colums. These are respectively a (0, α1 )-matrix and a (0, α2 )-matrix. The term obtained in the lower right hand corner is now row(M1 ) ⊗ row(M 2 ) ⊗ column(M1 ) ⊗ column(M2 )

(11.44) = row(M1 ) ⊗ row(M2 ) ⊗ column(M1 ) ⊗ column(M2 ) Now look at the overlapping shufﬂe product of two compositions β1 , β2 in a slightly different way. Intersperse both compositions with zeros in any way to obtain vectors of the same length n and such that the two zero-interspersed compositions when put on top of each other give a 2 × n matrix without zero columns. Take the column sum vector of this 2 × n matrix. Doing this in all possible ways gives the overlapping shufﬂe product (obviously). Now, all matrices M which yield given matrices M1 , M2 , are obtained from M1 , M2 by interspersing them with zero columns to have the same number of columns and such that when stacked on top of each other they give a matrix without zero columns. Combining these two remarks gives that for a given M1 , M2 the possible originating M precisely give the compositions column(M) that arise as terms of the overlapping shufﬂe product of column(M1 ) and column(M2 ), proving the commutativity of (11.43). For left distributivity a similar diagram must be commutative. it is obtained from the one at hand by replacing the lower left hand morphism by id ⊗ μ and the bottom morphism by m ⊗ id ⊗ id. It is a trivial matter to check that left distributivity does not hold; it already fails for any α = [a1 , a2 ] with a1 = a2 . If one switches ‘row’and ‘column’in (11.37) there results a second comultiplication morphism that is left distributive but not right distributive. One easily sees that on a symmetric sum of quasisymmetric monomials μP is commutative. Indeed if M is a (0, α) - matrix then row(M tr ) = column(M) and column(M tr ) = row(M) and the transpose M tr is a (0, α )-matrix for some permutation α of α. Given right distributivity of this second comultiplication on QSymm to check that it does indeed give the functorial multiplication of Witt vectors or power series it sufﬁces to check things on power series of the form 1 + at which is a triviality. 11.45. Co-unit for the second comultiplication on QSymm. There is also a counit morphism εP for μP . It is given by 1 for m ≤ 1 (11.46) εP ([a1 , a2 , · · · , am ]) = 0 for m ≥ 2

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377

To be a counit for μP it must satisfy the condition that the following composition of morphisms is the identify μP

εP ⊗ id

m

QSymm −→ QSymm ⊗ QSymm −→ QSymm ⊗ QSymm −→ QSymm and also the composition of morphisms obtained from this by switching εP and id for the middle arrow. Let α = [a1 , a2 , · · · , am ]. As εP is zero on anything of length 2 or more the only (0, α)-matrices that need to be considered have only one row. There is just one such matrix, viz (a1 , a2 , · · · , am ). This gives that the composed morphism in question is indeed the identity. 11.47. Second multiplication on NSymm. Dually the second comultiplication on QSymm gives a second mutliplication morphism (of coalgebras) on NSymm. Denoting this one by *, there is the right distributivity formula (which follows from the right distributivity of μP over μS in QSymm). Zα ∗ Zγi Zβ ∗ Zγi

where μ(Zγ ) = (Zα Zβ ) ∗ Zγ = Zγi ⊗ Zγi

i

i

This formula is also in [162]. The dual of the recipe for the second comultiplication given in (11.39) is due to Solomon, [373]. 11.48. Unit for the second multiplication on NSymm. The second multiplication, mP on NSymm deﬁnes a multiplication on each homogeneous component NSymmn . The unit for this second multiplication is the element Zn . So the ‘unit’ for the second comultiplication on all NSymm is the inﬁnite sum 1 + Z1 + Z2 + Z3 + · · · which does not live in NSymm but only in a suitable completion. The same situation holds for Symm. Here, for each n the second multiplication, sometimes called inner multiplication, turns each homogenous summand Symmn into a commutative ring with unit element hn , and the sum of all these elements, which does not live in Symm itself is the unit element for the second multiplication on all of Symm. So the autoduality of Symm is not prefect at this level. The second multiplication and second comultiplication are nicely dual to each other but the second unit only exists in a suitable completion while the second counit is present for Symm itself. This is understandable because the second multiplication and second comultiplication do not respect the grading. 12. Free, cofree and duality properties of Symm This short section concerns ways of obtaining Symm etc. by means of free and cofree constructions and consequences thereof. And open questions concerning these matters. All algebras and coalgebras in this section will come with a unit resp. counit element seen as.morphisms from and to Z.

378

M. Hazewinkel

12.1. Free algebras. To set the stage consider ﬁrst the well known case of free algebras. Let M be an Abelian group. A free algebra on M is an algebra Free(M) together with a morphism of Abelian groups iM : M −→ Free(M) such that the following universality property holds For every morphism of Abelian groups ϕ: M −→ A to a ring A there is a unique morphism of rings ϕ: ˜ Free(M) −→ A such that ϕi ˜ M =ϕ

(12.2)

This deﬁnes a functor Free: AbGroup −→ Ring that is left adjoint to the (forgetful) functor U: Ring −→ AbGroup that forgets about the multiplication. Pictorially, the situation looks as follows.

Free(M)

$1 ␸ A

iM

(12.3)

␸ M The symbol group ∃1 in the diagram above means ‘there exists a unique’. Such a functor ‘Free’ exists. It is given by the tensor algebra over M. 12.4. Tensor algebra and tensor coalgebra. For an Abelian group M let T i M = M ⊗i = M ⊗ M ⊗ · · · ⊗ M (and T 0 M = Z) be the i-fold tensor product. There are ∼ =

natural isomorphisms ψn,m : T n M ⊗ T n M −→ T n+m M. Now consider ∞ T n M, TM = n=0

iM : M −→ TM, eM : Z −→ TM,

(12.5)

pM : TM −→ M, εM : TM −→ Z

where the morphisms iM , pM are obtained by identifying M and T 1 M and eM , εM by identifying Z and T 0 M. There is a ring structure on TM given by ψm,n

T m M ⊗ T n M −→ T n+m M and iM as unit morphism.

(12.6)

This is the tensor algebra and it is the free algebra on M with respect to the (canonical) morphism iM . There is also a coalgebra structure on TM given by n

−1 −1 −1 −1 −1 (ψ0,n , ψ1,n−1 , ψ2,n−2 , · · · , ψn−1,1 , ψn,0 ): T n M −→ ⊕ T i M ⊗ T n−i M i=0

(12.7) and εM as counit morphism The two structures do not combine to give a bialgebra structure. Far from it. 12.8. The graded case. In the present context the important case is when everything in sight is graded and M is a free Abelian group.

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So let M be a positively graded Abelian group ∞

M = ⊕ Mi

(12.9)

i=1

Then TM is graded by giving a pure tensor x1 ⊗ · · · ⊗ xm , xi ∈ Mi1 degree i1 + · · · im . If no grading on M is speciﬁed it is treated as graded with all nonzero elements of degree 1. If now M is moreover free with homogeneous basis {Tu: u ∈ U},

degree(Tu ) = ru

(12.10)

Free(M) is ZTu: u ∈ I. For instance if the index set is N and the degree of Ti is i Free(M) = NSymm (as far as the underlying graded algebra is concerned). There is also a commutative version with Freecomm (M) being the maximal com∞

mutative qoutient of Free(M). So if M = ⊕ Zhi , degree (hi ) = i, Freecomm (M) = i=1

Symm.

12.11. Cofree coalgebras. All coalgebras in the following are supposed to come with a co-unit and morphisms of coalgebras are supposed to be compatible with the co-units involved. Given an Abelian group M the cofree coalgebra over M (if it exists) is a coalgebra CoFree(M) (over the integers) together with a morphism of Abelian groups πM : CoFree(M) −→ M with the following universal property: For every coalgebra C together with a morphism of Abelian groups ϕ : C −→ M there is a unique morphism of coalgebras ϕ˜ : C −→ CoFree(M) such that πM ϕ˜ = ϕ

(12.12)

Pictorially this looks as follows: CoFree(M)

'1

␸∼

␲M

C

(12.13)

␸ M

a picture that is completely dual to the one, see(12.3), for a free algebra over a module. The universality property (12.12), (12.13) says, given existence, that ‘CoFree’ is right adjoint to the forgetful functor which assigns to a coalgebra the underlying Abelian group. Whether the cofree coalgebra on47 an Abelian group always exists is unkown. It does do so in the case of free Abelian groups, their duals48 and certain other cases, see [194]. 47 It pays to be linguistically careful here: free algebra on an Abelian group; cofree coalgebra over an

Abelian group. 48 These are not free in the inﬁnite rank case.

380

M. Hazewinkel

Let M be (graded) free with (homogeneous) basis {Tu: u ∈ I}; and let C be a coalgebra ϕ together with a morphism of Abelian groups C −→ M. As an Abelian group the tensor coalgebra consists of all noncommutative polynomials in the Tu . This is not always large enough to serve as receiving object for the unique morphism ϕ˜ covering ϕ as required in (12.12), (12.13). Consider for example the coalgebra C = Z ⊕ Zx, μ(x) = x ⊗ x, ε(x) = 1 together with the morphism of Abelian groups that sends Z to zero and x to one of the basis elements T of M. It is obivous that the n-th component of the covering morphism, ϕ˜ n: C −→ T i M, must (always) be given by ϕ˜ n = ϕ⊗n μn where μn is the iterated comultiplication μ0 = ε, μ1 = id, μ2 = μ, · · · , μn+1 = (μ ⊗ id ⊗ · · · ⊗ id)μn , · · ·

(12.14)

Thus in the example at hand ϕ˜ n (x) = T ⊗ T ⊗ · · · ⊗ T (n factors), so that the ϕ(x) ˜ ‘is’ the power series 1 + T + T 2 + · · · + T n + · · · which is not in TM. To obtain the free coalgebra on M consider the completion Tˆ M of all formal noncommutative power series in the indeterminates Tu , u ∈ I. The free coalgebra over M is now something in between TM and Tˆ M of which the elements can be called for good (but varying) reasons ‘rational noncommutative power series’, ‘representative noncommutative power series’, ‘Schützenberger-recognizable noncommutative power series’, or ‘recursive noncommutative power series’. loc. cit. Things become a good deal easier in the graded case. In this case, i.e. for graded morphisms of graded coalgebras into graded Abelian groups, the tensor coalgebra is the cofree coalgebra. To sees this ﬁrst observe that a morphism of graded Abelian groups C −→ M takes the degree 0 component of C to zero. Next for any homogeneous x ∈ C of degree n, say, all terms in its terated coproduct μm (x) for m > n must contain a factor from C0 . Hence ϕ˜ m (x) = ϕ⊗m μm (x) = 0 for m > n and things are OK. There is also a cocommutative version where TM is replaced by its maximal cocommutative sub coalgebra of all symmetric tensors. In the case of graded free Abelian groups the graded free and graded cofree constructions are graded dual in that there is a natural isomorphism CoFree(M)dual = Free(M dual ) For the nongraded case there is also a duality but things are more complicated.49 12.15. Inheritance of structure. Now let C be a coalgebra and apply the free algebra construction to the augmentation module I = Ker(ε). This gives (in any case) an algebra which will still be denoted Free(C). It inherits a coalgebra structure as follows. By functoriality the coalgebra morphism gives an algebra morphism Free(μ)

Free(C) −→ Free(C ⊗ C)

(12.16)

49 Partly this comes from the fact that while the linear algebraic dual of a coalgebra is immediately an

algebra it is not true that the full dual of an algebra is a coalgebra; instead, one must take a submodle of the dual called the zero-dual.

Witt vectors. Part 1

381

Further there is the natural morphism ic ⊗ ic : C ⊗ C −→ Free(C) ⊗ Free(C) and by the freeness property this induces a morphism of algebras Free(C ⊗ C) −→ Free(C) ⊗ Free(C) which when composed with (12.16) gives the desired coalgebra structure morphism μFree(C)

Free(C) −→ Free(C) ⊗ Free(C)

(12.17)

By construction it is an algebra morphism and so Free(C) becomes a bialgebra. Dually, consider an algebra A and apply the cofree construction to the quotient module A/Z. Denote the result by CoFree(A). This one inherits an algebra structure in a similar way. Applying CoFree to the multiplication morphism gives a coalgebra morphism CoFree(A ⊗ A) −→ CoFree(A)

(12.18)

On the other hand there is a fairly obvious (canonical) morphism of Abelian groups CoFree(A) ⊗ CoFree(A) −→ (A ⊗ A)/Z that by the cofreeness property induces a morphism of coalgebras CoFree(A) ⊗ CoFree(A) → CoFree(A ⊗ A)

(12.19)

which when composed with with (12.18) gives the desired multiplication morphism which is by construction a coalgebra morphism so that the result is again a bialgebra. 12.20. Symm, NSymm and QSymm again. The cofree coalgebra over the integers is CoFree(Z) = Z ⊕ ZZ1 ⊕ ZZ2 ⊗ · · · μ(Zn) =

n i=0

Zi ⊗ Zn−i ,

where Z0 = 1, ε(Zn ) = δ0,n

(12.21)

It is now easy to see that NSymm = Free(CoFree(Z)), Symm = Freecomm (CoFree(Z))

(12.22)

It is far less easy (except via duality) to see in this way that QSymm = CoFree(Free(Z)), but it is interesting to do the exercise as this way the overlapping shufﬂe product comes about naturally. It is also not easy to see directly that Symm = CoFreecocomm (Free(Z)) (except, again, via duality. The duality argument is simple. QSymm = NSymmdual = Free(CoFree(Z))dual = CoFree(CoFree(Z)dual ) = CoFree(Free(Zdual )) = CoFree(Free(Z)) where ‘dual’ stands for graded dual.

382

M. Hazewinkel

13. Frobenius and Verschiebung and other endomorphisms of and the Witt vectors. After the rather abstract stuff of the two previous sections, it is a pleasure to return in this one to such concrete things as ghost components of power series and Witt vectors. There are a vast number of functorial operations on the functorial rings (A) ∼ = W(A) which is no surprise as every ring endomorphism of Symm induces such a functorial operation; and, as a free ring on countably many generators there are very many ring endomorphisms. Things become different if one also requires such things as preservation of the Abelian group structure on the rings W(A) or preservation of the full functorial ring structure or one requires other interesting properties. Probably the easiest to describe are the socalled Verschiebung operations. 13.1. Verschiebung. The Verschiebung50 operators are deﬁned on the (A) = 1 + tA[[t]] by Vn: (A) −→ (A),

a(t) → a(t n )

(13.2)

To see what this means at the ghost component level, apply t dtd log. Now if

t

d log a(t) = p1 + p2 t + p3 t 2 + · · ·, dt

d log a(t n ) = t(nt n−1 (p1 + p2 t n + p3 t + · · · )) dt

and so, on the ghost component level Verschiebung is given by npr/n if r is divisible by n V n pr = 0 if r is not divisible by n

(13.3)

This corresponds to the ring endomorphism of Symm (also denoted by Vn ) hr/n if r is divisible by n Vn: Symm −→ Symm, hr → 0 if r is not divisible by n (13.4) 13.5. Proposition. The Verschiebung morphisms deﬁne additive functorial endomorphisms of the (A). That is they are Hopf algebra endomorphisms of Symm. This is obvious from the deﬁnition. It can also be checked by the usual ghost component argument using (13.3). 50 The word means ‘shift’ and comes for the German. So it should indeed be written with an upper case ﬁrst letter. In the case of the p-adic Witt vectors it was introduced by Witt.

Witt vectors. Part 1

383

13.6. Frobenius operations. For the Frobenius operations on the (A) use again the symmetric function point of view. So again write the universal power series h(t) as

h(t) = 1 + h1 t + h2 t 2 + · · · = (1 − ξi t)−1 (13.7) i

and deﬁne fn h(t) =

i

(1 − ξin t)−1

(13.8)

Formally, of course, one observes that the right hand side of (13.8) is symmetric in ξ. So there are universal polynomials Qn,1 (h), Qn,2 (h), · · · ∈ Symm = Z[h1 , h2 , · · ·] for the coefﬁcients in the power series (13.8) and now deﬁne fn a(t) = 1 + Qn,1 (a)t + Qn,2 (a)t 2 + · · ·

(13.9)

The sequences of polynomials Qn (h) = (Qn,1 (h), Qn,2 (h), · · · ) that deﬁne the Frobenius operations are determined by wr (Qn (h)) = wnr (h)

(13.10)

In particular Qn,1 (h) = wn (h)

(13.11)

Qp,pr (h) only involves the variables h1 , hp , · · · , hpr+1

(13.12)

and

The higher Qn,r (h) are difﬁcult to write down explicitely. The Frobenius Hopf algebra endomorphisms of Symm corresponding to the Frobenius operations are (of course) given by hr → Qn,r (h). For later purposes it is useful to have a little explicit knowledge about these polynomials. 13.13. Lemma. The polynomials Qn,r (h) determining the Frobenius operations satisfy p Qp,pr (h) ≡ hpr mod p, for p a prime number (13.14) (13.15) Qn,r (h) ≡ nhrn mod (h1 , h2 , · · · , hrn−1 ) Both statements are proved by induction. From (13.11) it follows that (13.15) holds for r = 1. Suppose with induction that it holds for all 1 ≤ i < r. By the deﬁning property and this induction assumption wr (Qn ) ≡ rQn,r mod (h1 , h2 , · · · , hnr−1 ). On the other hand wnr (h) ≡ nrhnr mod (h1 , h2 , · · · , hnr−1 ). Comparing these two congruences proves (13.15). Further (13.14) holds for r = 0, again by (13.11). Suppose (13.14) holds for 0 ≤ i < r. It follows that pr−i pi Qp,pi ≡ pi hpi−1 pr−i+1 (13.16)

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and consequently the ﬁrst r terms of r

r−1

wpr (Qp ) = (Qp,1 )p + p(Qp,p )p

+ · · · + pr−1 (Qp,pr−1 )p + pr Qp,pr

Cancel mod pr with the ﬁrst r terms of r+1

wpr+1 (h) = (h1 )p leaving

r

r

+ p(hp )p + · · · pr−1 (hpr−1 )p + pr (hpr )p + pr+1 hpr=1

p pr Qp,pr ≡ pr hpr + pr+1 hpr+1

proving (13.14). In the same formal vein let ζn be a primitive n-th root of unity. Then always n

j=1

1 − ζnj t 1/n = (1 − t)

and so the Frobenius operation can also be written as n

a ζnj t 1/n fn a(t) =

(13.17)

j=1

The ghost components of h(t) as in (13.7) are the power sums in the ξ and so on the level of ghost components the Frobenius operations are characterized by fn pr = pnr

(13.18)

The Frobenius operations fp for p a prime number have a Frobenius like property: fp a(t) ≡ a(t)∗p mod p

(13.19)51

Here ∗p means taking the p-th power in the ring (A). The congruence (13.20) takes place in (A), so it means that there is a power series b(t) ∈ (A) such that b(t)p (fp a(t)) = a(t)∗p . To prove this ﬁrst observe that for power series of the form (1 − xt)−1 by the deﬁnition of Frobenius and product fp (1 − xt)−1 = (1 − xp t)−1 = ((1 − xt)−1 )∗p

(13.20)

Now in any commutative ring, including (A), (y + z)p ≡ yp + zp mod p and so from (13.20) it follows that (13.19) holds for any ﬁnite product power series (1 − ξ1 t)−1 · · · (1 − ξt)−1 51 Let K/Q be an unramiﬁed extension of the p-adic numbers with ring of integers A. Then the Galois p group Gal(K/Qp ) is cyclic with a generator σ. Let k be the residue ﬁeld of K. Then, as extensions of ﬁelds A/Zp = Wp∞ (k)/Wp∞ (Fp ) and σ is the Frobenius endomorphism of Wp∞ (k). It is characterized by σ(a) ≡ ap mod p. This has to do with the terminology employed here.

Witt vectors. Part 1

385

(and in fact under the right circumstances (such as here) also for inﬁnite products). Alternatively, for an arbitrary series take its Witt vector coordinate expression a(t) =

∞

1 − xd t d

−1

(13.21)

d=1

This stabilizes in that the ﬁrst n coefﬁcients of a(t) only depend on the ﬁrst n x-coordinates. As (13.20) already holds for all factors on the right of (13.21) one gets (13.19) in full generality by a simple limit argument. 13.22. Theorem. The fn :Symm −→ Symm induce functorial ring endomorphisms on the rings (A). They are also the only ring endomorphisms of Symm that do so. 13.23. Corollary. The only ring automorphism of Symm that preserves its coring object structure (as an object in the category of rings) is the identity. I.e. the object Symm is rigid as a coring object in the category CRing. Note that in contrast to the Liulevicius theorem 10.8, homogeneity is not needed. 13.24. Proof of the corollary. The Frobenius morphisms fn satisfy fn pr = pnr . So the only one that induces an operation that is always injective is f1 = id (as is seen by looking at the operation on any Q-algebra).52 13.25. Proof of theorem 13.22. For the ﬁrst statement of the theorem is sufﬁces to verify this on the ghost components, that is the power sums pr . On these the sum comultiplication, sum counit, product comultiplication, product counit are respectively given by μs (pr ) = 1 ⊗ pr + pr ⊗ 1, εs (pr ) = 0, μP (pr ) = pr ⊗ pr , εP (pr ) = 1

(13.26)

So the characterizing property (13.18) of the Frobenius morphism proves that they preserve the functorial ring structure on the (A). Now let ϕ be a ring endomorphism of Symm that respects the structures (13.26). In particular that means that ϕ must take primitives of the Hopf algebra Symm into primitives. The space of primitives of Symm consists of the linear (integer) combinations of the pr . So the image of pr under ϕ is of the form ϕ(pr ) = crj pj (13.27) 52 Incidentally, all Frobenius operations except the identity are also not injective on the integers. Indeed let p be a prime number and consider the vector a = (a1 , a2 , a3 , · · · ) ∈ ZN with an = p if n is not a multiple of p, and an = 0 if n is a multiple of p. Then by the ghost-vector criterium in 9.93 (with

A = Z and all the endomorphisms the identity) this vector is a ghost-Witt vector i.e. there is a Witt vector x ∈ W(Z) such that w(x) = a. For this vector fp (x) = 0.

386

M. Hazewinkel

Now use μP (ϕ(pr )) = (ϕ⊗ϕ)μP (pr ) = ϕ(pr )⊗ϕ(pr ) and 1 = ϕ(1) = ϕ(εp (pr )) = εp (ϕ(pr )) to ﬁnd that the coefﬁcients crj must satisfy crj crj = crj , crj crj = 0 if j = j , crj = 1 j

and this is only possible if all but one of these coefﬁcients are zero and that last one is equal to 1. Thus there is a mapping σ : N −→ N such that ϕ(pr ) = pσ(r)

(13.28)

Now consider the Newton relations (9.57): pr = rhr − (p1 hr−1 + p2 hr−2 + · · · + pr−1 h1 )

(13.29)

It easily follows with induction that pr ≡ (−1)r+1 hr1 mod(h2 , h3 , · · · )

(13.30)

Let n = σ(1) and suppose with induction that it has been shown that ϕ(pu ) = pnu , ϕ(hu ) ≡ 0 mod(h2 , h3 , · · · ) for r − 1 ≥ u ≥ 2

(13.31)

Now apply ϕ to the Newton relation (13.29) and use (13.28), (13.30) and the induction hypothesis (13.31) to ﬁnd that σ(r)

(−1)σ(r)+1 h1

≡ ϕ(pr ) (r−1)n )(−1)n+1 hn1 ) mod(h2 , h3 , · · · ) ≡ rϕ(hr ) − ((−1)(r−1)n+1 h1

and so σ(r) = nr and ϕ(hr ) ≡ 0 mod(h2 , h3 , · · · ). It follows that ϕ = fn . 13.32. Note that the proof does not use any integrality of coefﬁcients statement and so the theorem still holds over the rationals. Indeed it works over any ring and so the theorem is true for the coring object algebras SymmR = Symm ⊗ R over any ring R. 13.33. Frobenius endomorphisms of QSymm. The Frobenius morphisms fn on Symm extend to Frobenius endomorphisms on QSymm ⊃ Symm. The formula is simple fn ([a1 , a2 , · · ·, am ]) = [na1 , na2 , · · ·, nam ]

(13.34)

It is obvious from the original deﬁnitions of the ﬁrst and second comultiplication, and the description of the overlapping shufﬂe product that these are Hopf algebra endomorphisms that preserve the second comultiplication and second co-unit. They are also the only ring endomorphisms of QSymm that do so. To prove this consider a Hopf algebra endomorphism ϕ of QSymm that preserves the second comultiplication. First it must take primitives into primitives. The primitives are spanned by the compositions of length 1, see 11.36. Further μP ([a]) = [a] ⊗ [a],

εP ([a]) = 1

Witt vectors. Part 1

387

and so, exactly as above, there is a mapping σ: N −→ N such that ϕ([a]) = [σ(a)]

(13.35)

Let σ(1) = n and let J2 be the ideal in QSymm spanned by all compositions of length 2 or more. This is indeed an ideal. Now also εP ϕ = εP and it follows that ϕ(J2 ) ⊂ J2 . Further [r] ≡ [1]r modJ2 and so ϕ([r]) ≡ ϕ([1]r ) = ϕ([1])r = [n]r ≡ [nr] and so ϕ is equal to fn on the submodule spanned by the [a]. Now consider a composition of length 2. As ϕ respects the ﬁrst comultiplication μS (ϕ([a, b])) = (ϕ ⊗ ϕ)(1 ⊗ [a, b] + [a] ⊗ [b] + [a, b] ⊗ 1) = 1 ⊗ ϕ([a, b]) + [na] ⊗ [nb] + ϕ([a, b]) ⊗ 1

(13.36)

The term [na] ⊗ [nb] on the right of (13.36) can only come under μS from the composition [na, nb] and if there were any other compositions involved in ϕ([a, b]) that would show up after applying μS . So ϕ([a, b]) = [na, nb]. This argument easily continues. 13.37. Corollary. The Frobenius operators on Symm are given on the monomial symmetric functions by fn m(λ1 ,λ2 ,··· ,λm ) = m(nλ1 ,nλ2 ,··· ,nλm )

(13.38)

There are (of course) also other ways of seeing this, but this is a particularly elegant way. Thus on the monomial symmetric functions the Frobenius operators are easy to describe. It is far more difﬁcult to describe them on the hλ , though of course there are the messy formulas giving the h’s in terms of the p’s. Dually the Verschiebung operators are easy to describe on the h’s and difﬁcult to describe on the monomial symmetric functions. 13.39. Corollary. There are no second multiplication preserving Hopf algebra endomorphisms of NSymm that descend to a Frobenius endomorphism of Symm as a quotient of NSymm except the identity. Indeed such a morphism would involve some degree increasing part. Its dual would therefore involve some degree increasing part and be a second comultiplication preserving Hopf algebra endomorphism of QSymm, which is impossible by 13.25 above. This gives another negative answer, as conjectured, to a question posed in [200, 201]. 13.40. The Frobenius morphisms fp , for p a prime number also satisfy the Frobenius like property [a1 , a2 , · · · , am ]p ≡ [pa1 , pa2 , · · · , pam ] modp

(13.41)

This is more or less immediate from the deﬁnition of the overlapping shufﬂe product. Indeed, the terms of this p-th power are the column sums of all matrices without zero columns of which the rows are obtained from a1 , a2 , · · · , am by interspersing zeros. If all the rows are equal the column sum [pa1 , pa2 , · · · , pam ] results. If not all rows

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M. Hazewinkel

are equal any (non identity) cyclic permutation yields another different matrix of the same type (this uses that p is prime) and so all the other terms in the shufﬂe power occur with a coefﬁcient divisible by p. 13.42. ‘Multiplication by n’ operator. Consider the operator of adding (in (A) or W(A)) an element to itself n times, i.e. in (A) taking the n-th power of a power series. This operator is denoted [n] : (A) −→ (A), a(t) → a(t)n

(13.43)

This operator as an endomorphism of Symm is given by the composition of maps μn

mn

Symm −→ Symm⊗n −→ Symm

(13.44)

where μn is the n-fold ﬁrst comultiplication as deﬁned in (12.14) and mn is the n-fold multiplication. This composed map can be written down for any Hopf algebra H. In general it is not a Hopf algebra endomorphism, nor even an algebra or coalgebra morphism unless H is commutative and cocommutative. Then it is a Hopf algebra endomorphism. In spite of not being a Hopf algebra endomorphism in general these maps have proved to be most useful in various investigations in Hopf algebra theory, see e.g. [323]. 13.45. Homothety operations. Consider the Hopf algebra SymmR = Symm⊗R over a ring R. For every R-algebra A and every u ∈ R consider the operation ua(t) = a(ut), a(t) ∈ (A)

(13.46)

These are the homothety operators and they clearly deﬁne additive functorial Abelian group endomorphisms of the (A). The associated Hopf algebra endomorphism of SymmR is u(hn ) = un hn

(13.47)

As will be seen later the homothety, Verschiebung, and Frobenius endomorphisms together, in a very precise sense, generate all R-Hopf algebra endomorphisms of SymmR . There are quite a good many relations among all these operators. They are summed up in the following theorem. 13.48. Theorem. There are the following identiﬁes between operations on the functors R (−), WR (−), respectively the Hopf algebra SymmR . uu = uu

(13.49)

1 = f1 = V1 = id

(13.50)

Vm Vn = Vnm

(13.51)

fm fn = fmn

(13.52)

if gcd(m,n) = 1, then

fm Vn = Vn fm

(13.53)

Witt vectors. Part 1

fn Vn = [n]

389

(13.54)

Vn (a(t) ∗ fn b(t)) = (Vn a(t)) ∗ b(t), fn x, y = x, Vn y,

a(t), b(t) ∈ (A)

x, y ∈ Symm

(13.55) (13.56)

uVn = Vn un

(13.57)

fn u = un fn

(13.58)

u + v =

∞

Vn rn (u, v)fn

(13.59)

n=1

where the rd (X, Y ) are the integer coefﬁcient polynomials in two variables determined by (1 − Xt)(1 − Yt) =

∞

(1 − rd (X, Y )t d )

(13.60)

d=1

These last two, (13.59), (13.60) constitute of course the formula for the addition of Teichmüller representatives in Witt coordinates. Applying t dtd log to the two sides of (13.60) it follows that the polynomials rj (X, Y ) are determined by the relation Xn + Y n = drd (X, Y )n/d (13.61) d|n

For example r1 (X, Y ) = X + Y r2 (X, Y ) = −XY r3 (X, Y ) = −(X2 Y + XY 2 ) r4 (X, Y ) = −(X3 Y + 2X2 Y 2 + XY 3 ) r5 (X, Y ) = −(X4 Y + 2X3 Y 2 + 2X2 Y 3 + XY 4 ) r6 (X, Y ) = −(X5 Y + 3X4 Y 2 + 4X3 Y 3 + 3X2 Y 4 + XY 5 ) It immediately strikes one’s attention that the coefﬁcients in r2 , r3 · · · , r6 are all negative. This is true for all n ≥ 2 as an immediate consequence of the ReutenauerScharf-Thibon result 9.71. Another striking fact is that all monomials that possibly can occur do in fact occur with nonzero coefﬁcient. I know of no proof for that but deﬁnitely believe it to be true. 13.62. Caveat. All these formulas are written from the functorial operations point of view (except (13.56) where the only possible interpretation is in terms of endomorphisms). So, for instance, (13.54) means that for an element a(t) ∈ (A) there

390

M. Hazewinkel

is the equality fn (Vn (a(t))) = [n](a(t))

(13.63)

If U1 , U2 , U are functorial operations and u1 , u2 , u are the endomorphism that induce them, then seeing an element a(t) ∈ (A) as a morphism Symm −→ A (as must be done to have this correspondence between endomorphisms and operators), the relation beteen the operation U and the endomorphism u is U(a(t)) = a(t) ◦ u

(13.64)

where the small circle denotes composition. It follows that under the correspondence ‘functorial operation’ ⇔ ‘endomorphism’ the order of composition reverses. Indeed U1 (U2 (a(t))) = U1 (a(t) ◦ u2 ) = (a(t) ◦ u2 ) ◦ u1 = a(t) ◦ (u2 ◦ u1 ) 13.65. Proof of theorem 13.48. Most of these are pretty trivial using the ghost component formalism (see also 5.16). As before for any natural number r and element a(t) ∈ (A) let d log(a(t)) (13.66) dt Then the characterizing ghost component description of the various functorial operations are nsr/n (a(t)) if n divides r (13.67) sr (Vn a(t)) = 0 if n does not divide r sr (a(t)) = coefﬁcient of t r in t

sr (fn a(t)) = snr (a(t))

(13.68)

sr (ua(t)) = ur sr (a(t))

(13.69)

sr ([n]a(t)) = nsr (a(t))

(13.70)

So, using this, here is a proof of (13.54): sr (fn Vn a(t)) = srn (Vn a(t)) = nsr (a(t)) = sr ([n]a(t)) using (13.68), (13.67), (13.70) in the order named. And here is how (13.59) is tackled with the ghost component formalism nsr/n (rn (u, v) fn a(t)) if n divides r sr (Vn rn (u, v) fn a(t) = 0 if n does not divide r nrn (u, v)r/n sr/n (fn a(t)) if n divides r = 0 if n does not divide r r/n nrn (u, v) sr (a(t)) if n divides r = 0 if n does not divide r

Witt vectors. Part 1

391

and so ∞ sr Vn rn (u, v) fn (a(t)) = nrn (u, v)r/n sr (a(t)) = (un + vn )sr (a(t)) n|r

n=1

= sr (u a(t) + v a(t)) where of course, as indicated, the sum of elements in (A) is taken according to the addition law in the Abelian group (A). Here is the proof of (13.53) as a further illustration of ghost component techniques srm/n (a(t)) if n divides rm sr (fm Vn a(t)) = srm (Vn a(t)) = 0 otherwise and as (m, n) = 1 the divisibility condition in the formula above is the same as ‘n divides r’, which is what turns up when calculating sr (Vn fm a(t)). All the other statements of theorem 13.48 are proved the same way, except of course the duality statement (13.56). For this statement use the orthonormal dual pair of bases {hλ }, {mλ }, see (9.40), (9.42). Let κ = (κ1 , κ2 , · · · , κm ), λ = (λ1 , λ2 , · · · , λn ) be two partitions. Recall that h(κ1 /r,κ2 /r,··· ,κm /r) if all κj are divisible by r fr mλ = m(rλ1 ,rλ2 ,··· ,rλn ) , Vr hκ = 0 otherwise Thus

fr mλ , hκ =

1

if lg(λ) = lg(κ) and κi = rλi for all i

0

otherwise

mλ , Vr hκ =

(13.71)

1

if lg(λ) = lg(κ), r divides all κi and κi /r = λi for all i

0

otherwise (13.72)

comparing (13.71) and (13.72) now gives the duality formula (13.56) 13.73. Convention. Thus from (13.56) the Frobenius and Verschiebung endomorphisms of Symm = Z[h] are (graded) dual to each other. But Symm is also isomorphic to its graded dual. That makes it a triﬂe difﬁcult to agree what should be called Frobenius and what Verschiebung. The convention is that the degree nondecreasing endomorphisms (of these) are always called Frobenius and the degree nonincreasing ones Verschiebung. Both are Hopf algebra endomorphisms. Then the Frobenius endomorphisms always respect the second comultiplication (and the corresponding co-unit), but not the second multiplication, and thus induce functorial ring endomorphisms of the W(A) ∼ = (A) = CRing(Symm, A), while the Verschiebung endomorphisms respect the second multiplication (and corresponding unit), but not the second comultiplication, and thus induce ring endomorphisms on the rings of coalgebra morphisms CoAlg(C, Symm).

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M. Hazewinkel

13.74. Ring of Hopf endomorphisms of Symm. Naturally the next step is to try to determine the full ring of Hopf algebra endomorphisms of Symm, which is the same as the determination of all additive functorial operations on the functor . Consider again, see 12.11ff, the cofree coalgebra over Z, i.e. the coalgebra ∞ hi ⊗ hj , ε(hn ) = 0 for n ≥ 1 CoFree(Z) = ⊕ Zhn , h0 = 1, μ(hn ) = n=0

i+j=n

(13.75)

Now Symm = Z[h] is the free algebra on Cofree(Z), more precisely it is the free ∞

algebra over the augmentation submodule ⊕ Zhn ⊂ CoFree(Z) and hence a Hopf n=1

algebra endomorphism of Symm is the same thing as a morphism of coalgebras from CoFree(Z) to Symm that takes h0 to 1. Hopf(Symm, Symm) ∼ = CoAlg (CoFree(Z), Symm)

(13.76)

where the prime indicates that only morphisms that take h0 to 1 are permitted. In more pedestrian terms, as Symm = Z[h] is free on the generators hn an algebra endomorphism ϕ of Symm is the same thing as specifying a sequence of polynomials ϕ(hn ) and this sequence yields an endomorphism of Hopf algebras if and only if the sequence 1, ϕ(h1 ), φ(h2 ), ϕ(h3 ), · · · is a curve (= divided power sequence). The graded dual of CoFree(Z) is Z[T ] the ring of polynomials in one variable T . Thus, taking graded duals, and using that Symm and its graded dual are isomorphic as Hopf algebras (and even as coring objects in the category of rings and as ring objects in the category of coalgebras) one sees that CoAlg(CoFree(Z), Symm) ∼ = CRing(Symm,Z[T ])

(13.77)

and thus, tracing out what the prime in (13.68) means for the right hand side of (13.69), Hopf(Symm, Symm) ∼ = CRing (Symm, Z[t])

(13.78)

where this time the prime means that only morphisms are allowed that take the degee ≥1 part of Symm into the degree ≥1 part of Z[T ]. But his last object is easy to describe: an element of CRing (Symm, Z[t]) is simply an inﬁnite sequence of polynomials in T with constant terms zero. Things go exactly the same way for SymmA = Symm ⊗ A. As the isomorphism between Symm and its graded dual takes the hn into the en it is best to see this sequence of polynomials as the images of the en . Thus in this way a Hopf algebra endomorphism ϕ of SymmA is exactly the same thing as an inﬁnite × inﬁnite matrix with coefﬁcients in A ⎞ ⎛ a11 a12 a13 · · · ⎜ a21 a22 a23 · · · ⎟ ⎟ ⎜ Mϕ = ⎜ a31 a32 a33 · · · ⎟ ⎠ ⎝ .. .. .. .. . . . . such that in each row there are but ﬁnitely many coefﬁcients that are nonzero.

Witt vectors. Part 1

393

It remains to ﬁgure out how these matrices are to be added and multiplied (composed) when interpreted as encodings of Hopf endomorphisms of the Hopf algebra Symm. That will be the business of section 15 below. To do something similar for the characteristic p case as well (and the p-adic case) one needs to ﬁgure out how the p-adic Witt vectors ﬁt with the power series point of view. That is the subject matter of the next section. 14. Supernatural and other quotients of the big Witt vectors It is when trying to ﬁgure out how the p-adic Witt vectors ﬁt with the power series point of view of sections 9-13 that the Witt vector coordinates have a decided advantage. Recall that in terms of Witt vector coordinates, 9.63, as a set W(A) = {(x1 , x2 , x3 , · · · ): xn ∈ A}

(14.1)

and that two Witt vectors are added and multiplied by means of universal polynomials with integer coordinates x +W y = (μS,1 (x, y), μS,2 (x, y), μS,3 (x, y), · · · ) x ×W y = (μP,1 (x, y), μP,2 (x, y), μP,3 (x, y), · · · )

(14.2)

where x = (x1 , x2 , x3 , · · · ), y = (y1 , y2 , y3 , · · · ), and where the polynomials μS,i , μP,i are recursively given by wn (μS,1 (X; Y ), μS,2 (X; Y ), μS,3 (X; Y ), · · · ) = wn (X) + wn (Y ) wn (μP,1 (X; Y ), μP,2 (X; Y ), μP,3 (X; Y ), · · · ) = wn (X)wn (Y ) with wn (X) =

d|n

n/d

dXd

(14.3)

(14.4)

The important fact to notice is now that wn (X) depends only on the Xd for d a divisor of n, and hence that the n-th addition and multiplication polynomials μS,n , μP,n are polynomials that only involve the Xd and Yd with d a divisor of n. Thus, for instance, Z[X1 , Xp , Xp2 , Xp3 , · · · ], where p is a prime number, is a sub Hopf algebra and sub coring object of Z[X] = Z[X1 , X2 , X3 , · · · ], which means that it deﬁnes a quotient functor of W, which is manifestly Wp∞ , so that the p-adic Witt vectors are a functorial quotient of the big Witt vectors. There are lots more of such quotient functors. 14.5. Nests. A nest is a nonempty subset N of the natural numbers N such that together with any n ∈ N all the divisors of n are also in N. Some simple examples of nests are for any given ﬁxed n ∈ N, the set {1, 2, 3, · · · , n} p∞ = {1, p, p2 , p3 , · · · } for any ﬁxed n ∈ N, the set {d: d divides n}

(14.6)

Unions and intersections of nests are nests. Every nest contains the natural number 1.

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M. Hazewinkel

14.7. Supernatural numbers. A supernatural number is a formal expression of the form

n= pαp αp ∈ N ∪ {0, ∞} p

where the product is over all prime numbers p. This is exactly the same as specifying for every prime number p an element of the extended natural numbers {0} ∪ N ∪ {∞}. Given a supernatural number n there is a nest associated to it (often denoted with the same symbol), viz n = Nn = {m ∈ N: vp (m) ≤ αp for all prime numberp} where vp is the p-adic valuation on the integers. The last two examples in (14.6) are nests coming from a supernatural number. The ﬁrst example from (14.6) is not of this from (if n ≥ 3). 14.8. Supernatural quotients of the big Witt vectors. Let N be a nest. Let Symm (N) = Z[Xn : n ∈ N] ⊂ Symm. By the remarks made in the beginning of this section these form sub Hopf algebras of Symm and hence deﬁne quotient Witt vector functors WN (A) of the big Witt vectors, called the N-adic Witt vectors. The most important ones are Wp∞ (A), the p-adic Witt vectors Wpn (A), the p-adic Witt vectors of length n + 1 Wn (A), the Witt vectors of length n. The last named quotient functor is the one deﬁned by the next {1, 2, · · · , n}. This notation is not entirely consistent with the other two. However, the quotient Witt vectors deﬁned by the nest {d : d|n} are so seldom used (if ever) that this seems justiﬁed. Sometimes the curious appellation “nested Witt vectors” is used for the elements of a WN (A). 14.9. Operations on nested Witt vectors. A ﬁrst question now is now which of the functorial additive operations a, Vr , fr survive to deﬁne operations on the various quotients WN . Here of course these functorial operations are the ones deﬁned by transferring the operations denoted by the same symbols on (A). They are therefore characterized by wr/n if n divides r wr fn = wrn , wr Vn = , wr a = ar wr (14.10) 0 otherwise The answer to the above question of which operations ‘descend’ to operations on the various WN is fairly simple. The homothety operations a always deﬁne a quotient operation.

Witt vectors. Part 1

395

For the Verschiebung operations the situation is simple. The polynomials deﬁning Vn are either zero or of the form Xr/n for r a multiple of n. So by the nest property they always exist on the quotients WN . Of course many of them may be zero. The situation with the Frobenius operators is different and slightly more complicated. Let p be a prime number. The polynomials deﬁning fp are degree increasing by a factor of p and because fp is characterized by fp (pn ) = ppn at the ghost component level, the n-th polynomial deﬁning fp involves the indeterminate Xnp . See also lemma 13.13 above. So for fp to descend to a nest quotient in full generality, i.e. to exist for the functor WN on all of CRing, it is necessary and sufﬁcient that the nest contain together with any n ∈ N also all its p-power multiples pi n. In particular fp exists on the p-adic Witt vectors. However, if one restricts attention to the functor on rings of characteristic p the operation fp always exists on the quotients WN because in that case the Frobenius is given by raising each coordinate of a Witt vector to its p-th power See again lemma 13.13. 14.11. Möbius function. The next question is whether there exists lifts, that is an additive functor morphism WN −→ W which composed with the quotient map W −→ WN gives the identity on WN . This will involve the Möbius function from number theory and combinatorics. Explicitly this function is deﬁned by μ(1) = 1 μ(n) = (−1)r if n is the product of r different prime numbers

(14.12)

μ(n) = 0 if n is divisible by the square of a nontrivial prime number The function has the characterizing property μ(d) = 1

(14.13)

d|n

(which also deﬁnes it recursively). Let N(p) = {n ∈ N: (p, n) = 1} be the set of all natural numbers relatively prime to a given prime number p. It immediately follows from the above that 1 if n is a power of p μ(d) = (14.14) 0 otherwise d|n,d∈N(p) 14.15. Sectioning the projection W(A) −→ Wp∞ (A). To set the stage here is the abstract situation. Suppose there is an Abelian group M together with a surjective projection π: M −→ Mp to another group Mp . Suppose there is a section, i.e. a morphism of Abelian groups s: Mp −→ M such that πs = idMp . Then = sπ is an idempotent endomorphism of M with image s(Mp ) and kernel Ker(π). Further the projection π induces an isomorphism from (M) = s Mp to Mp . It is such an idempotent endomorphism that will now be

396

M. Hazewinkel

constructed in the case of the canonical projection πp : W(A) −→ Wp∞ (A)

(14.16)

in the case of Z(p) algebras, i.e. rings A in which all prime numbers except the give prime number p are invertible. To this end consider the operation p−typ =

μ(n) n

Vn fn (14.17)

This functorial operation satisﬁes wr if r is a power of p wr p−typ = 0 otherwise

(14.18)

(and it is characterized by this property, of course). This is an immediate consequence of (14.14) and (14.10). Note that it follows that p−typ is idempotent. (First for rings A of characteristic zero and then for all by functoriality). 14.19. Remark. p−typ commutes with the operations a, fp , Vp . 14.20. Comments. p−typ is not easy to write down explicitly, say, in terms of its deﬁning polynomials. It is deﬁnitely not something like murdering each coordinate that is not at a p-th power position. Easy explicit calculations show that it does not preserve the unit element and that it is not multiplicative. This already shows up at the second coordinate for all prime numbers larger than 2 and in the third coordinate for the prime number 2. 14.21. Proposition. For each Z(p) -algebra A, a Witt vector x = (x1 , x2 , x3 , · · ·) ∈ W(A) is in the image of p−typ (A): W(A) −→ W(A) if and only if fp x = 0 for all prime numbers p different from p. First, if fp x = 0 for all prime numbers different from p, then fn x = 0 for all n = N(p), n ≥ 2 and so p−typ x = [1]x = x. Second, if A is of characteristic zero and x ∈ W(A) is in the image of p−typ (A) then the characterizing property (14.18) shows that fp x = 0. Third, suppose that x ∈ W(A) is in the image p−typ . Choose any ˜ covering A. ˜ of x ∈ W(A) for some characteristic zero Z(p) -algebra A lift x˜ ∈ W(A) ˜ x is also a lift of x, and so fp p−typ x˜ = 0 implies Then, by idempotency, p−typ (A)˜ fp x = 0. 14.21.

Theorem. For Z(p) -algebras the canonical projection πp : W(A) −→ πp

Wp∞ (A) induces an isomorphism of Abelian groups p−typ (W(A)) −→ Wp∞ (A). Pictorially things are described by the following commutative diagram.

Witt vectors. Part 1

W(A)

W(A)

Fp–typ

397

␲p

∪ Fp–typ (W(A)) wpr

wpr

Wp` (A)

␲p

(14.22)

wpr

A It follows just about immediately that the theorem holds for A of characteristic zero. Now let A

W(Ã)

⌽p–typ

W(ψ)

W(A)

⌽p-typ (W(Ã))

␲p

Wp⬁(ψ)

W(ψ) ⌽p–typ

⌽p–typ (W(A))

Wp⬁(Ã)

␲p

(14.23)

Wp⬁(A)

˜ that covers it with be any Z(p) -algebra and choose a characteristic zero Z(p) -algebra A ˜ corresponding projection ψ: A −→ A and set J = Ker(ψ). Consider the commutative diagram (14.23) above. It follows immediately from the fact that the upper right πp is an isomorphism that the lower right πp is surjective. Now let x ∈ p−typ W(A), and take a y ∈ W(A) that ˜ that lifts it. Then x˜ = p−typ (˜y) is a lift of x. Now maps to x and take a y˜ ∈ W(A) suppose that πp (x) = 0 (lower right hand corner in the diagram). Then πp (˜x) has all its coordinates in J. By the commutativity of diagram (14.22) another, possibly ˜ that different, lift of x can be obtained by applying p−typ to the vector in W(A) has the same coordinates πp (˜x) at the p power spots and zeros elsewhere. Let this vector be x˜ . But the morphism p−typ is given by some universal polynomials with coefﬁcients in Z(p) and so all coordinates of x˜ are in J, and so x = W(ψ)˜x = 0. This proves that the lower πp in diagram (14.23) is also injective. 14.24. The morphism p−typ (transferred to the power series case) is an example of ‘p-typiﬁcation.’ This is general notion, due to Pierre Cartier, deﬁned for curves in any commutative formal group. The one involved here is the second simplest example, ˆ m . See [85] namely the one for the one dimensional multiplicative formal group G and [192], section 15.2. 14.25. Decomposition. It follows pretty directly from the preceding that over a Z(p) -algebra each Witt vector can be written uniquely as a sum

398

M. Hazewinkel

x=

Vn yn , yn ∈ p−typ W(A)

(14.26)

n∈N(p)

of shifted p-typical Witt vectors. And thus for Z(p) -algebras the functor of the big Witt vectors decomposes (as an Abelian group) into a direct product of N(p) copies of the p-adic Witt vectors. 14.27. Iterated nested Witt vectors. Let N and M be two nests. These can be multiplied to form a new nest MN = {mn : m ∈ M, n ∈ N} ⊂ N

(14.28)

There is now, in [28], the theorem (observation) that if M ∩ N = {1} there is a canonical functorial isomorphism WMN (A) −→ WM (WN (A)) which essentially come from the Artin-Hasse exponential W(−) −→ W(W(−)) which is the essential part of the cotriple structure on the functor of the big Witt vectors, see subsections 16.43 and 16.59 below. The case M = {1, p, p2 , p3 , · · · }, N = N(p) = (n ∈ N : (n, p) = 1}, so that MN = N can already be found, more or less, in [336]. This is a good, actually better, substitute for the decomposition (14.26), and it works also for rings that are not Z(p) -algebras 15. Cartier algebra and Dieudonné algebra Consider again the matter of Hopf algebra endomorphisms of Symm, or, equivalently, the matter of determining all functorial additive operations on the Witt vectors W(A) ∼ = (A). More generally consider the Hopf algebra SymmR over a ring R and consider the isomorphic functors and W on the category of R-algebras and the additive operations on them, or, equivalently, the Hopf R-algebra endomorphisms of SymmR . As was already indicated these can be desribed by some inﬁnity × inﬁnity matrix with only ﬁnitely many nonzero entries in each row. See 13.74 above. There are other ways to see such a thing and to describe all Hopf algebra endomorphisms over R. 15.1. (h − h)-matrix of a Hopf R-algebra endomorphism of SymmR . One other way is as follows. Because SymmR is free as an algebra over the ring R an algebra endomorphsim ϕ of it is uniquely speciﬁed by a sequence of polynomials, e.g. the images ϕ(hn ) of the free polynomial generators hn . To be a Hopf algebra endomorphism the sequence of polynomials d0 = 1, d1 = ϕ(h1 ) , d2 = ϕ(h2 ) , d3 = ϕ(h3 ), · · · must form a curve (divided power sequence) in SymmR , which means that εs (dn ) = 0 for n ≥ 1 and μs (dn ) = di ⊗ dj i+j=n

(15.2)

(15.3)

Witt vectors. Part 1

399

where μs and εs are the sum comultiplication and corresponding counit of SymmR . This means that the dn have constant terms zero and, I claim, are uniquely speciﬁed by the matrix (15.4) M(h−h) (ϕ) = ϕ(hi ), hj i,j I like to call this matrix the (h − h)-matrix of the endomorphism ϕ. There are only ﬁnitely many nonzero entries in each row because the ϕ(hi ) are polynomials (and hence bounded in weight). The proof of the claim is simplicity itself. By the autoduality of SymmR one has for example ! " ϕ(hn ), hk hl = ϕ(hi ) ⊗ ϕ(hj ), hk ⊗ hl i+j=n

=

ϕ(hi ), hk ϕ(hj ), hl

i+j=n

and thus, with induction, the (h − h)-matrix determines the inner products of the polynomials ϕ(hn ) with every monomial in the h’s and hence speciﬁes the polynomials themselves completely. One can also of course take any other free polynomial basis for SymmR , for instance the elementary symmetric functions, which gives the (h − e)-matrix of an endomorphism. It is perhaps interesting to calculate these matrices for some of the more important endomorphisms. For instance the Frobenius and Verschiebung endomorphisms. By deﬁnition h1 , e1 = 1 and hn , en = 0 for n ≥ 2. It follows with induction using the Newton relations, the fact that fn is a ring endomorphism, and duality, that pn , en = (−1)n+1 and hence that fn h1 , en = (−1)n+1 . Using again the Newton relations, ring morphism, and duality and induction one further ﬁnds fn hr , enr = 0. Or use fr hn , enr = hn , Vr enr = hn , en to see this. As all other entries of the matrix must be zero by degree considerations it follows that the (h − e)-matrix of the Frobenius Hopf algebra endomorphism fn of Symm has an entry (−1)n+1 at spot (1, n) and zeros everywhere else. Even easier one ﬁnds that the (h − e)-matrix of the n-th Verschiebung operator has a 1 at spot (n, 1) and zeros everywhere else. This is encouraging and suggests that every matrix of the type speciﬁed can arise. And this is indeed the case and can be proved this way. But this is not the easiest or most elegant way to see this. 15.5. The DE-matrix of a Hopf algebra endomorphism of SymmR . Let R[T ] be the algebra of polynomials in a single indeterminate over the ring R. The determining example for an additive operation on the functor R (the functor restricted to the category of R-algebras), and hence for the corresponding Hopf algebra endomorphism of SymmR , is the element (1 − Tt)−1 ∈ (R[T ])

(15.6)

400

M. Hazewinkel

in the sense that what a functorial operation does to this one example determines it completely. This claim will be examined a bit more closely further on. For the moment the argument is as follows. Let ϕ be an additive operation and suppose that ϕ((1 − Tt)−1 ) = 1 + a1 (T )t + a2 (T )t 2 + a3 (T )t 3 + · · ·

(15.7)

where the an (T ) are polynomials in T with coefﬁcients in R. Because the operation is supposed additive it must take the zero of the Abelian group (R[T ]), which is 1, into itself and so ai (0) = 0 for all i. Then the naive argument goes at follows. Take any element b(t) ∈ (A) for an R-algebra A. Write it formally as a product ∞

b(t) = 1 + b1 t + b21 t 2 + b3 t 3 + · · · = (1 − ηi t)−1 (15.8) i=1

Then ϕ(b(t)) =

∞

(1 + a1 (ηi )t + a2 (ηi )t 2 + a3 (ηi )t 3 + · · · )

(15.9)

i=1

This is more than a bit shaky. For one thing it is far from clear whether the product on the right hand side exists in a suitable sense. To make sense of things and also for other purposes rewrite the right hand side of 15.7 as a product

−1 ϕ((1 − Tt)−1 ) = 1 − cmn T n t m (15.10) m,n≥1

where for any m there are only ﬁnitely many n such that cmn = 0. This can be done in precisely one way. The ﬁniteness condition comes about precisely because the ai (T ) in (15.7) are polynomials (not power series for instance). The matrix MDE (ϕ) = (cmn )m,n

(15.11)

is the DE-matrix of ϕ.53 15.12. Proof of the DE principle (splitting principle). Manifestly, by the deﬁnition of the Frobenius, Verschiebung and homothety operations the right hand side of (15.10) is the operation Vm cmn fn (15.13) m,n

applied to the determining example power series (1 − Tt)−1 . It remains to show that (15.13) makes unique sense when applied to an arbitrary element b(t). To this end write b(t) in Witt vector coordinates as b(t) =

−1

(1 − xd t d )

d

53 DE stands for ‘determining example.’

(15.14)

Witt vectors. Part 1

401

Fix for the moment a power m0 of t. Because of the ﬁniteness condition there is an n0 such that cnm = 0 for all m ≤ m0 , n ≥ n0 . Because of the deﬁnition of Vm the ﬁrst m0 coefﬁcients of (15.13) applied to a ﬁnite product like (15.14) are determined by the part of (15.13) with m ≤ m0 . Now an fn is given by a series of polynomials of weights n, 2n, 3n, · · · . It follows that fn applied to an (1 − xt r )−1 with r > m0 n gives a result that has the ﬁrst m0 coefﬁcients (except the constant term 1) equal to zero. Thus to calculate the ﬁrst m0 coefﬁcients of (15.13) applied to (15.14) it sufﬁces to apply it to the ﬁnite product

(1 − xd t d )−1 =

<∞

(1 − ηi t)−1

(15.15)

i

d≤m0 n0

where the right hand side of (15.15) is also a ﬁnite product. And this is perfectly well deﬁned by additivity and functoriality. This proves the DE principle. This principle, apart form the inﬁnity questions just taken care of, is an algebraic analogue of the splitting principle in algebraic topology as used in, say, topological K-theory.54 15.16. Cartier algebra. By now it is clear that the ring of all additive operations on the functor is the set of all expressions (15.13) (with the stated ﬁniteness condition on the coefﬁcients involved). This ring, which is also the ring of Hopf R-algebra endomorphisms of the Hopf algebra SymmR , is called the Cartier algebra on R and denoted Cart(R).55 There are also already a number of calculating rules, as given by the formulas from theorem 13.48 viz formulas (13.49)–(13.54), (13.57)–(13.59). The remaining question is whether these sufﬁce. The most troublesome one (potentially) seems to be adding two expressions like (15.13). So consider a sum Vm bmn fn + Vm cmn fn (15.17) m,n

m,n

Manipulating these directly according to the rules given, which needs especially (13.51), just seems to lead to more and more sums at ﬁrst sight. More care is needed. To see how things work, calculate what the sum (15.17) perscribes for the ﬁrst few coefﬁcients, say 2. That means that all terms with a Vm , m ≥ 3 can be 54 Suppose one has an additive (in some suitable sense) operation in, say, topological K-theory. To ﬁnd out what it does to a given bundle over a space take a suitable covering space such that the induced buncle splits into a sum of line bundles. This can always be done. Thus, by functoriality and additivity it sufﬁces to know what the operation does to line bundles. And that in turn is governed (again by functoriality) by what it does to the universal line bundle over inﬁnite dimensional projective space. From this point of view I should have called (15.6) the universal example (overworking the word ‘universal’). The ‘principle’ that under suitable circumstances it sufﬁces to verify properties just for the case of line bundles is (or at least used to be) called the ‘veriﬁcation principle’; see e.g. [27]. 55 When seen as endomorphisms of the Hopf algebra Symm the elements of Cart(R) need to be written R in the form fn cmn Vm . See the ‘caveat’ 16.62 below. The need for the ﬁniteness condition is perhaps even clearer in this interpretation.

402

M. Hazewinkel

discarded leaving b1n fn + c1n fn + V2 b2n fn + V2 c2n fn n

n

n

(15.18)

n

Now b1n + cin = ∞ i=1 Vi ri (b1n , c1n fi and so the V1 part of the sum (15.17) is given by b1n + c1n fn n

(which is a ﬁnite sum). What is left is a sum V2 r2 (b1n , c1n ) f2 + V2 b2n fn + V2 c2n fn n

n

and so the V2 part of (15.17) is given by V2 b21 + c21 f1 + V2 (r2 (b12 , c12 ) + b22 + c22 ) f2 +

V2 (b2n + c2n ) fn

n≥3

Continuing this way (which is tedious) one sees that the calculating rules given sufﬁce to deal with sums. Composing operations means dealing with products like Vm b fn Vr c fs

(15.19)

Let d be the greatest common divisor of n and r. Then the product (15.19) is equal to Vm b fn Vr c fs = Vm b fn/d fd Vd Vr/d c fs = Vm b fn/d [d]Vr/d c fs = [d]Vm b fn/d Vr/d c fs = [d]Vm b Vr/d fn/d c fs = [d]Vmr/d br/d cn/d fsn/d So, back to (ﬁnite) sums again. The only remaining thing to check is that no inﬁnite sums turn up when multiplying (composing) two things like (15.13) which is immediate. So the given calculating rules sufﬁce. 15.20. Witt vectors as endomorphisms. Quite generally if is a unital-commutativering-valued functor on CRing to itself its restriction to R-algebras has (R) as part of its ring of additive endo operations. Indeed an R-algebra structure on a ring A is the same thing as a morphism of rings R −→ A, which on applying yields a morphism of rings (R) −→ (A) which, in turn, for each x ∈ (R) an associated additive operation (x, a) → xa.56 56 If the functor takes noncommutative rings as values, or more generally groups with an extra multiplication structure as values, there are two such operations, one on each side and these are additive if and only if resp. left distributivity or right distributivity holds for the extra multiplication structure over the group structure. This applies for instance to the case of the functor deﬁned by the ring of quasi-symmetric functions. QSymm, see [200].

Witt vectors. Part 1

403

So in the present case of the big Witt vectors W(R) must be part of the ring af ∼ additive endo operations of WR = R . It remains to identify these Witt vectors among the m,n Vm cmn fn . This is not difﬁcult. Inside the algebras of operations on R there are the operations of the form ∞

Vn xn fn , xn ∈ R

(15.21)

n=1

which take the determining example (1 − Tt)−1 into the power series ∞

(1 − xn T n t n )−1

(15.22)

n=1

which is a power series with constant term 1 in the variable Tt in Witt coordinate form. As addition of operations goes pointwise, the power series attached to the sum of two elements of the form (15.21) is the product of the corresponding power series (15.22). It follows that the special operations (15.21) constitute a subgroup of all operations that is (isomorphic to) the additve group of the Witt vectors W(R). Now consider the composition of an operation (15.21) with an operation of the form η with associated power series (1 − ηTt)−1 = η(1 − Tt)−1 . Appling η to (15.22) by deﬁnition gives ∞

(1 − xn ηn T n t n )−1

n=1

By the deﬁnition of the multiplication of Witt vectors, see (9.16) taking a product of a Witt vector a(tT ) with one of the form (1 − ηTt)−1 is the same as substituting ηtT for tT . Thus composition of operations and Witt vector multiplication agree in this case. As should be the case for the operation of η to ﬁt with the algebra structure (R) −→ (A). By additivity and distributivity (both of Witt vector multiplication over Witt vector addition and operation composition over operation addition) it folllows that a product of Witt vectors of the form ∞

n n −1 n n −1 (1 − xn T t ) (1 − xt T ) ∗ n=1

corresponds exactly to the composition of the corresponding operations. A degree argument now ﬁnishes the proof that multiplication of Witt vectors corresponds to the composition of operations of the form (15.21). 15.23. Theorem. The Witt vectors W(R) under the correpondence (15.21)–(15.22) form a functorial subring of the rings of operations End(R ) ∼ = End R (SymmR ). A somewhat remarkable fact in this case is that in the description 15.16 the subring of operations corresponding to (R) comes in Witt vector coordinates.

404

M. Hazewinkel

Still another way to see that the Witt vectors are part of Cart(R) will be brieﬂy discussed further below. 15.24. Further description of the Cartier algebra. It is now natural to describe the ring of operations Cart(R) as an overring of W(R), [84]. Let H be the skew polynomial ring over W(R) in commuting indeterminates fp , p a prime number, with the commutation relation with Witt vectors given by fp x = xfp fp where xfp = x(p) denotes the Witt vector arising from applying the p-th Frobenius operation to the Witt vector x. Next take power series in the commuting indeterminates Vp with commutation relations xVp = Vp xfp

and fp Vp = Vp fp

for

p = p

Let this ring be S. In it take the ideal J generated by the elements fp Vp − p,

Vp xfp − xVp ,

x ∈ W(R),

p a prime number

Then Cart(R) = S/J. 15.25. The ring of additive endo operations of Wp∞ . The situation in the case of the ring valued functor of the p-adic Witt vectors is a bit different. There is no ‘determining example’ (as far as I can see) like in the case of the big Witt vectors and it is not true that the operations V = Vp , f = fp (the only surviving Frobenius and Verschiebung operations) together with the homothety operations sufﬁce to generate all of them. In fact for p = 2 these are not even enough to produce the operation [2] of adding a p-adic Witt vector to itself. However, as above in 15.20 there are the p-adic Witt vectors Wp∞ (Z) as operations on Wp∞ and together with V, f these do sufﬁce to generate all additive operations. This is the topic of the next few pages. In this case I ﬁnd it more convenient to work in terms of Hopf algebra endomorphisms of the representing Hopf algebra of the p-adic Witt vectors. This is the Hopf algebra WH (p) = Z [X0 , X1 , X2 , · · ·]

(15.26)57

with addition and multiplication comultiplications determined by the p-adic Witt polynomials as in section 5 above. For this part of this section 15, temporarily, the notations of section 5 will be used again. 15.27. Endomorphisms of the inﬁnite dimensional additive group. To start with consider the inﬁnite dimensional additive group. This is the functor A → AN , represented by the ring Z[U] = Z[U0 , U1 , U2 , · · · ]

(15.28)

with sum comultiplication and product comultiplication and corresponding counits μs (Un ) = 1 ⊗ Un + Un ⊗ 1, εs (Un ) = 0, μp (Un ) = Un ⊗ Un , εp (Un ) = 1 (15.29) 57 The “WH” here stands for “Witt-Hopf” (as a sort of mnemonic).

Witt vectors. Part 1

405

AHopf algebra endomorphism of this Hopf algebra is given by a series of polynomials with coefﬁcients in the integers G(U ) = (G0 (U ), G1 (U ), G2 (U ), · · · ) which are additive. This means that Gn U1 + U1 , U2 + U2 , U3 + U3 , · · · = Gn (U1 , U2 , U3 , · · · ) + Gn U1 , U2 , U3 , · · ·

(15.30)

There are over a ring of characteristic zero, such as Z, not many such polynomials. The only ones are the linear ones, i.e. polynomials of the form Gn (U ) =

<∞

an,i Ui ,

an,i ∈ Z

(15.31)

i

where, as indicated, the sums are ﬁnite. 15.32. Determination of the Hopf algebra endomorphisms of WH (p) . Now let ϕ be a Hopf algebra endomorphism of WH (p) . It is given by a series of polynomials (ϕ0 , ϕ1 , ϕ2 , · · · ). These, when substituted in the polynomials w0 , w1 , w2 , · · · deﬁne an Hopf algebra endomorphism of the inﬁnite additive group and so are linear in the wi (X). Thus a series of polynomials (ϕ0 , ϕ1 , ϕ2 , · · · ) deﬁnes a Hopf algebra endomorphism of WH (p) if and only if wn (ϕ(X)) =

∞

an,i wi (X)

(15.33)

i=0

for suitable integers an,i of which only ﬁnitely many are nonzero for each n. Note that the integrality of the coefﬁcients an,i follows from (15.33). Just look at the coefﬁcients of the powers of X0 on the left and right hand side of (15.33). It also follows that the right hand side ‘comes from’ an integral vector over WH (p) . This ring is of the type described in the ‘ghost component integrality lemma’ 9.93. The ring endomorphism p ϕp is the one that takes each Xi to Xi . Here we are dealing with p-adic Witt vectors so the only denominators involved are powers of the prime number p. So the integrality criterion simpliﬁes to the single sequence for the prime number p. So in the case at hand ∞ i=0

an,i wi (Xp ) ≡

∞

an+1,i wi (X) modpn+1

(15.34)

i=0

is necessary and sufﬁcient for the sequence ϕ(X) as determined by (15.33) to deﬁne a Hopf algebra endomorphism of WH (p) and all Hopf algebra endomorphisms of WH (p) arise (uniquely) in this way.

406

M. Hazewinkel pj

Looking at the coefﬁcients of the powers X0 it follows from (15.34) that the integers an,i satisfy am,0 ≡ 0 mod(pm ),

m = 1, 2, 3, · · ·

am,i ≡ am+1,i+1 mod(pm+1 ),

m, i ∈ N ∪ {0}

(15.35)

(and these congruences are also sufﬁcient for (15.34) to hold). Now consider the following vectors over the integers bm = (p−m am,0 , p−m am+1,1 , p−m am+2,2 , · · · ), cn = (a0,n , a1,n+1 , a2,n+2 , a3,n+3 , · · · )

m = 1, 2, 3, · · ·

n = 0, 1, 2, 3, · · ·

(15.36)

By (15.35) these satisfy the criterium of lemma (9.93) for being a p-adic ghost Witt vector. So there are Witt vectors xm , yn ∈ Wp∞ (Z) such that

w(xm ) = bm ,

w(yn ) = cn

(15.37)

By deﬁnition the endomorphism f takes wr (X) into wr+1 (X) and the endomorphism V takes wr (X) into pwr−1 (X) and by the deﬁnition of the multiplication of Witt vectors w(x ·Wp∞ (X0 , X1 , X2 , · · · ) = (w0 (x)w0 (X), w1 (x)w1 (X), w2 (x)w2 (X), · · · ) for any x ∈ Wp∞ (Z). So the endomorphism ∞

n

f cn + c 0 +

n=1

∞

bm Vm

(15.38)

m=1

takes wr (X) into ∞

n

f ar,n+r wr (X) + ar,r wr (X) +

n=1

r

p

−m

m

ar,r−m p wr−m (X) =

m=1

∞

ar,i wi (X)

i=1

exactly as in (15.33). Because a Witt vector x ∈ Wp∞ (Z) starts with j zeroes, i.e lies in the ideal Vj Wp∞ (Z), if and only if its associated ghost vector starts with j zeroes (as also follows from the ghost vector criterium of lemma 9.93), the ﬁniteness condition of 15.33) translates into the statement that for each r all but ﬁnitely many of the Witt vectors cn in (15.38) lie in the ideal Vr Wp∞ (Z). Thus the ring of Hopf algebra endomorphisms of WH (p) consists of all expressions (15.38) with the ﬁniteness condition just stated and with the calculating rules Vf = p,

Vx = x(p) V,

xf = fx(p)

(15.39)

Witt vectors. Part 1

407

where x(p) denotes the result of applying the operation f to the p-adic Witt vector x and where p = [p] is the p-fold sum of the unit element in Wp∞ (Z). 15.40. The same technique can be used to determine the ring of endomorphisms of the Hopf algebra Symm as the representing Hopf algebra of the big Witt vectors, giving another way to obtain the ring CartZ . 15.41. The techniques of (15.32) as a method to determine the endomorphisms of (p) the Hopf algebra WHR break down when R is not a ring of characteristic zero for two reasons. One is that ghost component calculations as used are then not determining and second because in characteristic p > 0 there are other endomorphism of the additive groups than the linear ones; e.g. raising to the p-th power. Still I believe that the same picture holds. One reason is that the picture holds for (p) the WHR when k is a perfect ﬁeld of characteristic p > 0. This will be described below. However, the proofs are quite different. Still every endomorphism in this case does come from one deﬁned in characteristic zero (after the fact, i.e. after they have been determined in characteristic p). So there still appears to be a bit of work to do. 15.42. Dieudonné algebras. Let k be a perfect ﬁeld of characteristic p > 0. The Dieudonné algebra Dk consists of all expressions <∞

f n cn + c0 +

n=1

<∞

bm Vm ,

cn , bm ∈ Wp∞ (k)

(15.43)

m=1

subject to the calculating rules Vf = fV = p,

Vx = x(p) V,

xf = fx(p)

(15.44)

Recall that in this case the Frobenius operation on Wp∞ (k) is given by p p p p x(p) = x0 , x1 , x2 , x3 , · · · and hence is an isomorphism, so that in (15.43) the Frobenius and Verschiebung symbols can be written on the left or the right as convenient.Also recall that (see 13.13) (p) the sequence of polynomials that deﬁne the Frobenius endomorphism on WKk is p

p

p

X0 , X1 , X2 , · · · so that these endomorphisms in characteristic p take care of the ‘raising to the power p’ additive operations. One result in the present case is now (see [105], Chapter V, §1, No 3, p. 550): 15.45. Theorem. The endomorphim ring (over k) of the sub Hopf algebra k[X0 , (p) X1 , · · · Xn−1 ] ⊂ WHk is isomorphic to the ring Dk /Dk Vn . 15.46. Hirzebruch polynomials. (Very partial symmetric function formularium (3)). Consider again an additive (functorial) operation on the groups of Witt vectors (A).

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M. Hazewinkel

Such an operation is given uniquely by a series of polynomials 1 = K0 , K1 , K2 , · · · ∈ Z[e] = Z[e1 , e2 , e3 , · · · ]

(15.47)

such that μs (Kn ) =

n

Ki ⊗ Kn−i , where μs (en ) =

i=0

n

ei ⊗ en−i

(15.48)

i=0

is the sum comultiplication on Symm = Z[h] = Z[e].58 If the Kn are also homogeneous of degree n (where ei has degree i this is precisely what is deﬁned in [213], Chapter I, §1, p. 9 as a multiplicative sequence. The operation ψK deﬁned by the multiplicative sequence (15.47) is of course ψK : a(t) = 1 + a1 t + a2 t 2 + · · · + → 1 + K1 (a)t + K2 (a)t 2 + · · ·

(15.49)

A multiplicative sequence is uniquely determined by what in loc. cit. is called its characteristic power series Q(z) = 1 + K1 (1, 0, 0, · · · )z + K2 (1, 0, 0, · · · )z2 +

(15.50)

which can be seen as the value of the operation on the determining example 1 + zt because by the homogeneity condition on the Kn ψK (1 + zt) = Q(zt)

(15.51)

The homogeneity condition is also clearly equivalent to the property that in the DE formalism of (15.5) the DE-matrix of the operation is diagonal and so the operation on (A) in question is multiplication by a single Witt vector from (Z) (or (Q) if rational coefﬁcients are allowed in the K (and in the characteristic power series). This single particular Witt vector is Q(−t)−1 . The multiplicative sequence itself is recovered by considering the product ∞

Q(ξi t)

i=1

For each n the coefﬁcient of t n is a homogeneous symmetric polynomial in the ξ and hence a homogeneous polynomial of weight n in the ei . These are the original multiplicative sequence polynomials by multiplicativity of the sequence or, equivalently, additivity of the operation deﬁned by them. Inversely, start with any power series Q(t) over the integers or rationals and form the sum and product sQ (t) =

∞ i=1

Q(ξi t),

pQ (t) =

∞

Q(ξi t)

(15.52)

i=1

58 Elementary symmetric functions are used here in order to conform with usage in algebraic topology

such as in [213] where these things were ﬁrst introduced.

Witt vectors. Part 1

409

where for the sum case it is assumed that Q(0) = 0 and for the product case Q(0) = 1. The coefﬁcients of t n in (15.52) are symmetric in the ξ and hence are polynomials in the elementary symmetric functions. These are the n-th additive and multiplicative Hirzebruch polynomials deﬁned by the power series Q(t). For example the Todd genus operation is deﬁned by the power series ∞

Q(z) =

z 1 Bn 2n = 1 + (−1)n z + z 1 − e−z 2 (2n)!

(15.53)

n=1

and the Todd genus of a vector bundle is obtained by applying this operation of the total Chern class 1 + c1 (V )t + c2 (V )t 2 + · · · + cn (V )t n of a complex vector bundle of dimension n. The additive Hirzebruch polynomials serve to deﬁne functorial additive operations from (A) to AN . An example is the Chern character of a vector bundle deﬁned by applying the operation associated to the power series ez to the total Chern class of a vector bundle.59 15.54. Open problem. In subsection 15.5 (on the DE principle), formula 15.7 to be precise, an endomorphism of the Hopf algebra is given by an element of (Z[T ]), the Witt vectors over the ring of polynomials in one indeterminate. One wonders how the multiplication of these as Witt vectors ﬁts with the other bits of structure (composition and addition of endomorphisms). 16. More operations on the and W functors: λ-rings. There can be other operations than additive ones on an (Abelian) group valued functor that are important. This happens e.g. in algebraic topology and algebraic geometry with the operations on various functors induced by exterior powers (or, equivalently, symmetric powers) of vectorbundles. There are such functorial operations on the big Witt vectors. Indeed the Witt vectors (A) are functorial λ-rings.60 In this setting the Witt vectors have three more universality properties (that are interrelated). 16.1. Third universality property of the Witt vectors. The functor : CRing −→ λ− Ring is right adjoint to the forgetful functor the other way. This π means the following. First there is a canonical projection (A) −→ A (which is in fact the functorial morphism s1 (see (9.8), the ﬁrst ghost component morphism) and second there is the universal property (that says that (A) is the cofree object n 59 Here Q(0) = 1 = 0 and so a ﬁnite factorization 1 + c (V )t + · · · + c (V )t n = (1 + β t) is used. n i 1 i=1 60 I do not know whether the notation ‘’ for the functor A → 1 + tA[[t]] has anything to do with the

fact that the 1 + tA[[t]] are functorial λ-rings. I believe not. However, is also a standard notation for Symm the representing object of this functor.

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M. Hazewinkel

over A) to the effect that for every morphism of rings ϕ : S −→ A from a λ-ring to A there is a unique lift, that is a morphism of λ-rings ϕ˜ : S −→ (A) such that πϕ˜ = ϕ as illustrated in the diagram below.61

⌳(A) ᭚1 ∼ ␸ S

␸

␲ A

16.2. Fourth universality property of the Witt vectors. Symm, the representing object of the functor , is the free λ-ring on one generator. 16.3. Fifth universality property of the Witt vectors. There is a comonad (= cotriple) structure on and the coalgebras for this comonad are precisely the λ-rings. 16.4. Deﬁnition of λ-rings. A λ-ring is a unital commutative ring that comes equipped with a an extra collection of operations λi : A −→ A,

i = 1, 2, 3, · · ·

(16.5)

which ‘behave like exterior powers.’ In particular they are not additive for i ≥ 2. The phrase ‘behave like exterior powers’ means like the exterior powers of vector bundles over a space or like the exterior powers of representations of a group. Thus the ﬁrst requirement is that λn (a + b) = λn (a) +

n−1 i=1

λi (a)λn−i (b) + λn (b) =

n

λi (a)λn−i (b)

(16.6)

i=0

λ0 (a)

where by deﬁnition = 1 for all a ∈ A. In addition the exterior product operations are supposed to satisfy some properties on products and when iterated in the sense that there are some universal polynomials which give λn (xy) and λm (λn (x)) in terms of the λi (x), λj (y). These polynomials are speciﬁed by deﬁning exterior products on the rings (A) and declaring these particular rings with these particular exterior products to be λ-rings.62 That goes as follows. 61 This property rather nicely takes care of the complaint on page 1 of [263]. 62 Lambda rings were introduced by Alexandre Grothendieck in the course of his investigations into

Riemann-Roch type theorems, [48, 178, 179]. In spite of its attractive sounding title the reader is advised to steer clear of [148]. This book contains a great deal of most interesting material; it also contains nasty mistakes (as in the second paragraph of p. 15) and is sloppily written; see also the review by K R Coombes in Math. Rev. (88h:14011). In the older literature, see e.g. loc. cit. and [27, 239] a lambda ring is a ring with exterior product operations satisfying just the additivity condition (16.6) and the term ‘special lambda ring’ was used for what here is called a lambda ring.

Witt vectors. Part 1

411

A morphism of λ-rings is a morphism of rings that respects the exterior product structures, i.e. a unital morphims of rings ϕ : A −→ B such that ϕ(λiA (x)) = λiB (ϕ(x))

(16.7)

The exterior product structure on the (A) is deﬁned by

λm a(t) =

1 − ξi1 ξi2 · · · ξim t

−1

when

a(t) =

i1
∞

(1 − ξi t)−1

(16.8)

i=1

By deﬁnition a λ-ring is a unital commutative ring A equipped with exterior product operations λi such that λ1 = id and the mapping A −→ (A) given by x → (1 − λ1 (x)t + λ2 (x)t 2 − λ3 (x)t 3 + λ4 (x)t 4 · · · )−1

(16.9)

is a morphism of λ-rings (where (A) is given the exterior product structure (16.8)). Writing λ1 (x) = 1 + λ1 (x)t + λ2 (x)t 2 + λ3 (x)t 3 + · · ·

(16.10)

the second part of (16.9) says that x → λ−t (x)−1 , A −→ (A)

(16.11)63

is a morphism of λ-rings,while the ﬁrst part says that s1 λ−1 −t = id. Of course of make the whole thing consistent it has to be proved that (A) with the exterior product as speciﬁed by (16.8) is a λ-ring as deﬁned via (16.9). That is it has to be proved that (A) −→ ((A))

(16.12)

is a morphism of λ-rings. This will be done below, mixing in a number of lemmas which will also be useful later. 16.14. λ-rings vs σ-rings. Given a ring with exterior product operations λn deﬁne the corresponding symmetric power operations σ n by the formula n (−1)i λi (x)σ n−1 (x) = 0, where σ 0 (x) = 1 = λ0 (x)

(16.15)

i=0

63 There are a couple minus signs here compared to the deﬁnition as given in e.g. [239, 324]. The reason for that is that in the present text (for good reasons) the ring structure on (A) is speciﬁed by the ‘Hadamard like product’ (1 − xt)−1 ∗ (1 − yt)−1 = (1 − xyt)−1 with unit element (1 − t)−1 instead of (1 + xt) ∗ (1 + yt) = 1 + xyt with unit element 1 + t (as in [239, 324]). There are more good reasons for the particular choice made here (Which is also the choice made in the ﬁrst treatments of this subject, [84, 86]) For instance, as already mentioned, it works out nicer for the autoduality of Symm). Still another good reason is that the multiplication (1 − xt)−1 ∗ (1 − yt)−1 = (1 − xyt)−1 is how zeta functions of varieties over a ﬁnite ﬁeld multiply in the sense that ζ(X × Y ) = ζ(X) ∗ ζ(Y ), see [263], p. 2.

412

M. Hazewinkel

If the exterior product operations satisfy (16.6) then so do the symmetric product operations. Also, obviously a morphism of rings respects exterior products if and only if it respects symmetric products. Write σt (x) = 1 + σ1 (x)t + σ2 (x)t 2 + σ3 (x)t 3 + · · ·

(16.16)

then by deﬁnition a ring is a σ-ring if and only if σt : x → σt (x),

A −→ (A)

(16.17)

is a morphism of σ-rings. The relation between the λn and σ m can be succintly written σt (x) = (λ−t (x))−1

(16.18)

and so σt is precisely the mapping that ﬁgures in condition (16.11). Thus a σ-ring is exactly the same as a λ-ring. But the formulation of the property is just a tiny bit more elegant in terms of σ-rings (no minus signs). It is a nice little exercise to show that the σ-ring structure on (A) is given more explicitly by

−1 m −1 (1 − ξi t) (16.19) = σ 1 − ξi1 ξi2 · · · ξim t i

i1 ≤i2 ≤...≤im

This is in fact the Wronski relations again between elementary and complete symmetric functions (when the complete symmetric functions are written out as the sum of all monomial symmetric functions of the same weight). 16.20. Adams operations. Given a ring with exterior product structure operations λn the associated Adams operations,64,65 are deﬁned by ∞ n=1

n (x)t n = −t

d d log(λ−t (x)) = t log(σt (x)) dt dt

(16.21)

where the last equality of course follows from (16.18). Here also the formulation is just a bit more elegant in terms of the σ-structure.

64 The operations are named after J Frank Adams who ﬁrst deﬁned them in algebraic topology in the

context of complex vector bundles and topological K-theory, [14]. 65 Consider a Hopf algebra H and let μ and m be the n-th iterates of its comultiplication and muln n tiplication. Then the composite is the additive map (in general nothing more) [n], the n-fold sum of the identity (under convolution). In some of the literature, e.g. [141, 167, 270, 323] these maps are called Adams operations. That is a bit unfortunate as it does not ﬁt in e.g. the case of Symm. For one thing if the Hopf algebra is graded these maps [n] are homogeneous (of degree zero) while the Addams operations are degree increasing. Also it does not ﬁt with the case where the Adams operations came from, the cohomology of the classifying space BU. This does not mean that these maps are not important. They are, see loc. cit.

Witt vectors. Part 1

413

16.22. Theorem. The Adams operations on the rings (A) with exterior product operations given by (16.8) or, equivalently, with symmetric product operations given by (16.19), are the Frobenius operations. I like to call this observation the ﬁrst ‘Adams = Frobenius’ theorem. The proof is easy. First take a power series of the form x = (1 − at)−1 . Then λ1 (x) = x, λi (x) = 0 for i ≥ 2 and hence λ−u (x) = 1 − xu, −u

d xu log(λ−u (x)) = = xu + x2 u2 + x3 u3 + · · · du 1 − xu

And so n((1 − at)−1 ) = ((1 − at)−1 )∗n = (1 − an t)−1 = fn ((1 − at)−1 ) by the deﬁnition of product and Frobenius operations on (A). Now write an arbitrary element from (A) (formally) in the form

(1 − ξi t)−1, i.e. x = x1 + x2 + x3 + · · ·, xi = (1 − ξi t)−1 x= i

˜ The same calculation as above gives in terms of the addition in a (A).

∗n (1 − ξi t)−1 n (x) = x1n + x2n + x3n + · · · = i

−1 1 − ξin t = = fn x

66

i

16.23. Also observe that by the deﬁnitions the Frobenius operations on the (A), and hence the Adams operations, commute with the exterior and symmetric power operations. 16.34. Existence lemma ( for exterior power (and other) structures), [192], lemma 17.6.8, p.138. Let A be a characteristic zero ring with ring endomorphisms ϕn :A −→ A for all n ∈ N such that ϕ1 = id, ϕm ϕn = ϕmn and such that ϕp (a) ≡ ap mod(pA) for all prime numbers p and a ∈ A. Then there exists a unique mapping DA :A −→ (A) (resp. DA : A −→ W(A)) such that sn DA = ϕn (resp. wn DA = ϕn ). Moreover, this mapping is a ring morphism. This is an immediate consequence of the ghost Witt vector integrality lemma 9.93. 16.35. Lemma. Let α: A −→ B be a ring morphism and let both rings have exterior n , n . Then if αn = n α for products λnA , λnB with associated Adams operators A B B A n n all n, also αλA = λB α. This comes about because the Adams operations and the exterior products are related by (universal) polynomials with rational coefﬁcients. 66 Here, again, the splitting principle is used together with what has been called the ‘veriﬁcation principle,’ which says that under suitable circumstances it sufﬁces to verify things for line bundles.

414

M. Hazewinkel

Indeed these polynomials had better be given explicitly as they will also be needed further on. They are as follows ⎛

λ1

⎜ 2 ⎜ 2λ ⎜ ⎜ n = det ⎜ ⎜ 3λ3 ⎜ ⎜ .. ⎝ . nλn ⎛ 1 ⎜ ⎜2 ⎜ ⎜ ⎜ n!λn = det ⎜3 ⎜ ⎜ . ⎜ . ⎝ . n

1

0

λ1

1 .. . .. .

λ2 .. . λn−1

···

1

0

1

2 .. .

2 .. . n−1

..

··· .. . .. . λ1 λ2 ··· .. . ..

.

.

1

···

2

⎞ 0 .. ⎟ .⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ ⎟ 1⎠

(16.40)

λ1 ⎞ 0 .. ⎟ . ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ n − 1⎠ 1

(16.41)

Some care must be taken in reading these formulas. E.g. in λ2 = 12 (1 )2 − 1 2 1 2 1 1 1 2 2 , ( ) (x) must not be read as ( (x)) but as ( (x)) . These determinental formulas are exactly the same as those linking power sums and elementary symmetric functions in symmetric function theory ([281], p. 28), which is as must be because the deﬁning formulas are the same in the two cases.67 16.42. Clarence Wilkerson theorem. Let A be as above in lemma 16.34. Then there is a unique λ-ring structure on A such that the associated Adams operators are the given ring morphisms ϕn . Moreover the thus deﬁned exterior powers commute with the given morphisms ϕn . Proof. By the existence lemma 16.34 there is a unique ring morphism σt : A −→ (A) such that sn σt = ϕn for all n. This deﬁnes the exterior powers. Now observe that for all a ∈ A sm σt (ϕn (a)) = ϕm (ϕn (a)) = ϕmn (a) and on the other hand, using the deﬁning property of the Frobenius operation on the Witt vectors sm fn (σt (a)) = smn (σt (a)) = ϕmn (a) 67 It is because of this that the Adams operations are often called ‘power operations’.

Witt vectors. Part 1

415

Because A is of characteristic zero (so that s : (A) −→ AN is injective) this implies that fn σt = σt ϕn . Thus by lemma 16.35 σt respects the exterior powers and thus is a morphism of λ-rings as required. 16.43. Corollary. The rings of Witt vectors (A) are functorial λ-rings. Proof. First suppose that A is of characteristic zero. Then so is (A). Now let AH be the mapping (A) −→ ((A)) deﬁned by the given exterior product structure on (A). At this stage almost nothing is known about what properties it has. However, by deﬁnition of the Adams operations these satisfy sn,(A) ◦ AH = n . The operations n on (A) have been shown to be the Frobenius operations (the ﬁrst Adams = Frobenius theorem) and these Frobenius operations satisfy the conditions of the existence lemma 16.34. So the uniqueness part says that AH is equal to the morphism whose existence is guaranteed by that lemma and that one is a ring morphism by that lemma and respects exterior powers by theorem 16.42. This proves the corollary for characteristic zero rings. For an arbitrary ring take any characteristic zero cover. Everything in sight is functorial and so a little diagram chasing gives the result also in this case. 16.44. Remark. The following diagram commutes for any λ-ring.

L(A)

fn

L(A)

st A

st Cn

A

For instance because fn and n are given by the same polynomials in the exterior products (see (16.40) and the σt are morphisms of λ-rings. In fact more generally if A α and B are λ-rings and A −→ B is a morphism of λ-rings then the morphism commutes with the respective Adams operations. It follows from the commutativity of the above diagram that m n = mn and that the n respect the exterior powers. 16.45. Remark. The notation ‘AH’ for the functorial morphism of λ-rings (A) −→ ((A)), which could also be properly denoted σt,(A) , stands for ‘ArtinHasse’. It is in fact a variant of the Artin Hasse exponential68 in algebraic number 68 For the deﬁnition of the Artin-Hasse exponential in algebraic number theory and some of its uses see e.g. [412] to start with.

416

M. Hazewinkel

theory. More precisely if one takes a ﬁnite ﬁeld k, identiﬁes (k) = W(k), takes the quotient W(k) −→ Wp∞ , and sees the latter ring as the unramiﬁed mixed characteristic complete discrete valuation ring with residue ﬁeld k one ﬁnds a morphism of rings Wp∞ (k) −→ (Wp∞ (k)) which is in fact the classical Artin-Hasse exponential, see [192], sections 17.5, 17.6 and E2. By its deﬁnition the Artin-Hasse exponential AH is such that the left hand diagram below commutes. It follows that the right hand diagram also commutes.

⌳(⌳(A))

⌳(⌳(A))

AH

AH sn, ⌳(A)

⌳(A) fn

⌳(A)

⌳(sn, A)

⌳(A) fn

⌳(A)

16.46. Remark. The simplest example of a (nontrivial) λ-ring is probably the ring of integers with the exterior power structure λt (x) = (1 + t)x . So in this case λt is a kind of exponential, and that is a good way to think about it. In fact there are situations where the presence of an exterior power structure does lead to exponential type isomorphisms between the underlying ‘additive’ group of a ring and a multiplicative group of units of it, see e.g. [27], [388, 389]. 16.47. The essence of a λ-ring structure. Let A be a λ-ring. Then it comes with with a morphism of λ-rings σt : A −→ (A), σt (x) = (1 − λ1 (x)t + λ2 (x)t 2 − λ3 (x)t 3 + · · · )−1 . That means that there are formulas for the exterior product λn (x + y) of a sum of elements of A because σt is additive; there are formulas for the exterior product λn (xy) of a product of elements of A because σt is multiplicative; and there are formulas for the iterations of exterior products λm (λn (x)) because σt is respects exterior products (as a morphism of λ-rings). These formulas are given in terms of the addition, multiplication, and exterior products on the Witt vector ring (A). But these formulas are universal;69 they do not in any way depend on the ring A; they are certain polynomials with integer coefﬁcients determined by certain manipulations with symmetric functions over the integers. Thus there are formulas for λn (x + y), λn (xy), λm (λn (x)) as polynomials in the i λ (x) and λj (y) and, which is the real essence of the story, these polynomials have their coefﬁcients in the integers and are the same for all λ-rings A. These polynomials can be calculated. Either by working directly with the one universal example of the Witt vectors 1 + h1 (ξ)t + h2 (ξ)t 2 + h3 (ξ)t 3 + · · · and 1 + h1 (η)t + h2 (η)t 2 + h3 (η)t 3 + · · · over Z[h(ξ); h(η)] ⊂ Z[ξ; η], or, which I ﬁnd easier, by using the determinantal formulas (16.40) and (16.41). 69 Overworking that unhappy word again; but I know of none other that meets the case.

Witt vectors. Part 1

417

For example 1 1 3 1 1 ( ) (xy) − (1 2 )(xy) + (3 )(xy) 6 2 3 1 1 1 3 1 3 1 1 = (x) (y) − (x)1 (y)2 (x)2 (y) + 3 (x)3 (y) 6 2 3

λ3 (xy) =

= λ1 (x)λ2 (x)λ1 (y)λ2 (y) + λ1 (x)3 λ3 (y) + λ3 (x)λ1 (y)3 − 3λ1 (x)λ2 (x)λ1 (y)3 − 3λ1 (x)3 λ1 (y)λ2 (y) + 3λ3 (x)λ3 (y) 1 1 2 2 1 ( ) (λ (x)) − 2 (λ2 (x)) 2 2 1 2 2 1 2 1 1 2 1 2 = λ (x) − ( (x) − (x)) 2 2 2 2 1 2 2 1 2 2 1 4 = λ (x) − (x) + (x) 2 4 4

λ2 (λ2 (x)) =

= λ1 (x)λ3 (x) − λ4 (x) 16.48. -rings. A -ring is a ring together with a family of ring endomorphisms (n )n ∈ N such that 1 = id, m n = mn

(16.49)

I shall say that A is a ‘-ring with Frobenius morphism like property’ if moreover p (a) = ap modp

for all prime numbers p

(16.50)

Then if A is a λ-ring the associated Adams operations it becomes a -ring by what was remarked in 16.44 above. It is moreover a -ring with Frobenius morphism like property. To see this it sufﬁces to remark that that the Adams operations are given in terms of the symmetric powers (resp. the exterior powers) by the same polynomials that relate the power sum symmetric functions to the complete symmetric functions (resp. p the elementary symmetric functions). Now hp ≡ h1 modp and the statement follows. Another way to get it is to look at the deﬁning equation for p in Witt coordinates:

W(A) ␴⬘t A

wp Ψp

A

where σt is σt followed by the isomorphism eA : (A) −→ W(A). Because w1 σt = id, σt (a) is of the form (a, x2 , x3 , · · · ) and so p (a) = wp (a, x2 , x3 , · · · ) = ap + pxp ≡ ap modp

418

M. Hazewinkel

16.51. Thus for a characteristic zero ring the statements “A is a λ-ring’ and ‘A is a -ring with Frobenius morphism like property’ are equivalent. If A is not of characteristic zero this need not be the case. 16.52. Another result in this setting is that if A is a characteristic zero ring with exterior power operations deﬁned on it such that the associated Adams operations turn it into a -ring then it is a λ-ring. Note the differences between the two theorems 16.51 and 16.52. Thus the difference between -rings and λ-rings is an integrality matter (and such matters can be important). The idea of a -ring can easily be extended to a noncommutative setting (see below) and dualized. In view of the autoduality of Symm there should be some interesting notion of what a ‘co-λ-coring’ could be. Some sort of co--coring with an extra integrality property. 16.53. There are a number of noncommutative rings in various parts of algebra that behave almost like -rings and λ-rings. Thus it makes sense to develop a theory of noncommutative -rings just like it was (and is) important to have a theory of noncommutative symmetric functions; the theory of NSymm brieﬂy alluded to above in Section 11. The beginnings of a theory of noncommutative -rings have been established, see [325] (and [324] for a brief account). And it turns out that this theory relates nicely to the theory of noncommutative symmetric functions, loc. cit. The next topic is the comonad structure on the functor of the Witt vectors. 16.54. Monads and comonads. A monad70 in a category C is a triple (T, m, e) consisting of an endofunctor T : C −→ C, a natural transformation m: T T −→ T (where T T stands for the iterate of T , i.e. T T(A) = T(T(A))) and a natural transformation e: id −→ T such that the following diagrams commute.

TTT

Tm

TT

mT

TT

m m

T

T

Te

TT m

id

eT

T

id

T (16.55)

70 Other names for monad that are used (or have been used) in the literature are ‘triad,’ ‘standard construction,’ ‘fundamental construction,’ ‘triple’. Monads ﬁrst arose in work of Roger Godement, [171], in connection with the construction of simplicial objects and (standard) resolutions in connection with sheaf cohomology.

Witt vectors. Part 1

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Here ‘Tm’ means ‘m’ ﬁrst and then ‘T ’, i.e. at an object C take the morphism mC : T(C) −→ TT(C) = T(T(C)) and then apply the functor T to it to obtain a morphism that is also often denoted T(mC ); on the other hand ‘mT ’ at an object C, i.e. ‘T ’ ﬁrst and then ‘m’, is ‘m’ at the object T(C), often written mT(C) . An algebra (in the category C) for a monad (T, m, e), also called a T -algebra, is an object A ∈ C together with a morphism α: TA −→ A such that the following diagram commutes

TTA

T␣

␣

mA

TA

TA

␣

A

eA id

(16.56)

A

The opposite notions, i.e. the same notions in the opposite category, are those of a comonad and a coalgebra. Explicitly: a comonad (T, μ, ε) in a category C is an endo functor T of C together with a morphism of functors μ: T −→ T T and a morphism of functors ε: T −→ id such that (T μ)μ = (μT )μ,

(εT )μ = id = (T ε)μ

(16.57)

And a coalgebra for the comonad (T, μ, ε) is an object in the category C together with a morphism σ: C −→ TC such that εC σ = id,

(T(σ)σ = (μTC )σ

(16.58)

The equations (16.57) and (16.58) can of course be written out in diagram form, the diagrams involved being the ones one gets by reversing all arrows in (16.55) and (16.56). This seems hardly worth the ink and paper in view of the fact that they will shortly appear explicitly in the case of the particular comonad of the Witt vectors and its coalgebras, which are precisely the λ-rings (or, better, σ-rings). A standard text on monads and comonads is [33]; for a thorough up-to-date account see [284].71 16.59. Comonad structure on the Witt vectors. The claim is that the Witt vector functor (or W) is a comonad in the category CRing, with the comonad morphism given by the Artin-Hasse exponential and the ﬁrst ghost component morphism and that the coalgebras for this comonad are precisely the λ-rings. To prove this it must be shown that for each ring A the following diagrams (16.60) are commutative and that an exterior product structure σt : A −→ (A) gives a 71 And a, in my opinion, inspiring account.

420

M. Hazewinkel

λ-ring structure if and only if the diagrams (16.61) below are commutative.

⌳(AHA)

⌳⌳(A)

⌳⌳⌳(A)

⌳(A)

AH⌳(A)

AHA AHA

⌳(A)

⌳(s1, A)

⌳⌳(A)

s1,⌳(A)

AHA

id

⌳⌳(A)

⌳(A)

id

⌳(A) (16.60)

A

s1

⌳(A)

⌳(␴t)

␴t

id

A

⌳⌳(A) ␴t,⌳(A)

␴t

(16.61)

⌳(A)

This looks like there is a fair amount of work to do. Especially the triple iterate at the upper right hand corner of the left diagram in (16.60) could seem a bit intimidating. Actually everything has already been proved. Start with the commutativity of the right hand part of diagram (16.61). This is just saying (in a slightly fancy way) that σt is a morphism of λ-rings (with the fact that it is a ring morphism coming from the requirement that things take place in the category of rings). The left hand part of (16.61) just says that λ1 = id. As AH is the morphism determined by the exterior power structure on (A) (see the proof of 16.43) the commutativity of the left diagram of (16.60) is just proving that (A) is a λ-ring. The remainder of (16.60) is part of the deﬁnition of AH (right half ) and a consequence of that (see 16.45). The right half of the right diagram of (16.60) is also the ﬁrst part of the requirement that (A) be a λ-ring. 16.62. Theorem. Cofreeness of the Witt vectors. The λ-ring (A) of Witt vectors over A, together with the ﬁrst ghost component morphism s1 : (A) −→ A is the cofree λ-ring over A. This means that if S is a λ-ring and α: S −→ A is a morphisms of rings (i.e. forget the exterior product structure on S for the moment), then there is a unique morphism of λ-rings α˜ : S −→ (A) that lifts α. This is illustrated in the following diagram.

⌳(A) ᭚1 ∼ ␣ S

␣

s1 A

(16.63)

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By now this is simple to prove. First remark that if

(1 − ξi t)−1 ∈ (A) a(t) = 1 + a1 t + a2 t 2 + a3 t 3 + · · · = i

then by the deﬁnition of the λ-ring structure on (A) (see (16.8) λn a(t) ≡ 1 + en (ξ)t ≡ 1 + (−1)n+1 (hn (ξ) + P(h1 (ξ), · · · , hn−1 (ξ)) t

mod t 2

for some polynomial P with integer coefﬁcients. And so s1 λn a(t) = (−1)n+1 an + P (a1 , · · · , an−1 )

(16.64)

Now let α : S −→ A be a morphism of rings. Deﬁne α˜ : S −→ (A) as the composiσt

(α)

tion S −→ (S) −−→ (A). Then this α˜ does the job, and, with induction, it is also the only possibility because it is required to be a morphism of λ-rings and (16.64).72 16.65. λ-ring structure on Symm itself. There is a very simple and obvious λ-ring structure on the ring Z[ξ] = Z [ξ1 , ξ2 , ξ3 , · · · ] of polynomials over the integers, viz the one determined by λ1 (ξi ) = ξi ,

λj (ξi ) = 0

if j ≥ 2

(16.66)

The easiest way to prove that there is indeed such a λ-ring structure is to consider the ring endomorphisms given by n (ξi ) = ξin

(16.67)

These ring endomorphisms satisfy all the requirements of the Wilkerson theorem 16.42 and so there is a λ-ring structure on Z[ξ] for which the ring endomorphisms (16.67) are the associated Adams operations. The relations between the Adams operations and the exterior product operations are the usual ones linking power sums and elementary symmetric functions. In particular ⎛ 1 ⎞ 1 0 ··· 0 ⎜ .. ⎟ .. ⎜ 2 ⎟ . 1 2 . ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ .. .. 2 n!λn = det ⎜ 3 (16.68) . . 0 ⎟ ⎜ ⎟ ⎜ . ⎟ .. .. ⎜ . ⎟ . 1 n − 1⎠ . ⎝ . n

n−1

···

2

1

and it follows immediately that the λ-operations (exterior powers) is casu are indeed those of (16.66). 72 Again the proof would be more elegant if things were formulated for σ-rings.

422

M. Hazewinkel

Now consider the subring of the symmetric functions Z[h] ⊂ Z[ξ]. Obviously this subring is stable under the ring endomorphisms (16.67), so applying the Wilkerson theorem 16.42 again there is a λ-ring structure on Z[h] for which the morphisms (16.66) are the associated Adams operations which is obviously the restriction of the λ-ring structure (16.65) to the symmetric functions. 16.69. λ-ring structure on QSymm. In between the symmetric functions and Z[ξ] there sit the quasi-symmetric functions Z[h] = Symm ⊂ QSymm ⊂ Z[ξ]. Obviously, QSymm is stable under the ring morphisms (16.67) and so there is a corresponding λ-ring structure on QSymm, which is the restriction of the one on Z[ξ] and which extends the one just discussed on Symm. The associated Adams operations are the ring morphisms (16.67) which are the same as the Frobenius endomorphisms of 13.33. 16.70. Lyndon words. The proper tails of a word α = [a1 , a2 , · · · , am ], ai ∈ N, over the natural numbers are the words [ai , ai+1 , · · · , am ], 1 < i ≤ m. A word is Lyndon if it is lexicographically smaller than each of its proper tails. For instance [1, 3, 2] is Lyndon and [2, 1, 1] and [1, 1] are not Lyndon. A word α is primitive if the greatest common divisor of its entries is 1. 16.71. Theorem, [195, 198] (Free polynomial generators for QSymm over the integers). The elements λn α, n ∈ N, α primitive word, form a free polynomial basis over the integers of the ring QSymm. This has been a vexing problem since 1972 which was ﬁnally solved in 1999, see [193]. QSymm is a strong candidate for a theory of noncommutative Witt vectors (as a tool for classifying noncommutative formal groups amoung other things). 16.72. Discussion. The many different (?) operations on Symm. There are by now some ﬁve potentially different unary operations on Symm and it is perhaps wise to list them. (a) The exterior product operations that deﬁne the λ-ring structure on Symm. These are not additive of course. (b) The corresponding Adams operations as described by (16.66) (c) The rings (A) are functorial λ-rings. The functor A → (A) is represented by Symm: (A) = CRing(Symm, A). The functorial exterior product operations on (A) must therefore come from ring endomorphisms of Symm. These endomorphism cannot be Hopf algebra endomorphisms (because otherwise the functorial exterior product operations on W(A) would be additive. (d) The rings (A) have functorial Adams operations. These also must come from ring endomorphisms of Symm. Moreover these must be Hopf algebra endomorphisms and even coring object morphisms in the category of rings. (e) The Frobenius endomorphisms as deﬁned in subsection 13.6. These are the ones that induce the functorial Frobenius operarions on the (A).

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The ﬁrst Adams = Frobenius theorem, 16.22, says that the morphisms (d) and (e) are the same. By the remarks already made the only one which could conceivable also be the same is (b). And this is indeed the case 16.73. Theorem. (Second Adams = Frobenius theorem). The Adams operations n coming from the λ-ring structure on Symm are the same as the endomorphisms fn of Symm that induce the Frobenius endomorphisms of the functor (A) of the big Witt vectors. Proof. This follows immediately from the fact that fn (pm ) = pnm compared with (16.66). There is of course still more structure on Symm. Symm being autodual there are also all the duals of (a)–(e). It is not evident what all these dual operations are. 16.74. Theorem. Universal λ-ring on one generator. The ring of symmetric polynomials with the λ-ring structure deﬁned above is the universal λ-ring on one generator.73 This means the following. For each λ-ring A and element a ∈ A there is a unique morphism of λ-rings ϕ : Symm −→ A such that ϕ(e1 ) = a. Proof. As is easily veriﬁed from e.g. (16.58) the λ-ring structure on Symm satisﬁes λn (e1 ) = en . It follows that the only ring morphism that could possibly work is deﬁned by ϕ: en → λn (a). And there is such a ring morphism because Symm is free on the en . That this is actually a morphism of λ-rings requires a bit more work, as follows. Let x be an element of Symm, i.e. a polynomial in the elementary symmetric functions x = Px (e1 , e2 , · · · ) ≡ Px λ1 (e1 ), λ2 (e1 ), · · · Now consider λn (x). Because composition of lambda operations, a lambda operation applied to a product, and a lambda operation applied to a sum, are given by ‘universal polynomials’, that means the same polynomials for any λ-ring, see 16.47, there is a universal polynomial Qn,Px (with coefﬁcients in the integers)74 such that for any λ-ring and any element a in it λn(Px (λ1 (a), λ2 (a), · · · )) = Qn,Px (λ1 (a), λ2 (a), · · ·) Also ϕ is a ring homomorphism and so commutes with polynomials ϕ(Q(x1 , x2 , · · ·)) = Q(ϕ(x1 ), ϕ(x2 ), · · ·) 73 There is a far reaching generalization. As will be discussed below Symm is isomorphic to RS the direct sum of the representation rings of the symmetric groups. For a ﬁxed ﬁnite group G let G ×wr Sn be the wreath product of G and Sn . The direct sum of the representation rings R(G ×wr Sn ) is the free lambda ring on the irreducible representations of G as generators, [280]. 74 This polynomial is in fact the plethysm e ◦ P , see subsection 16.76 below. n x

424

M. Hazewinkel

It follows that ϕ commutes with the lambda operations, so that it is a morphism of λ-rings. 16.75. Comonadability vs cofreeness vs representability vs freeness. The various notions that passed under review in the previous few subsections are far from being unrelated. The following matters were discussed. (a) The functor : CRing −→ λ − Ring takes a ring A into the cofree λ-ring over A. (b) The functor comes with a comonad structure and the coalgebras of this comad are precisely the λ-rings. (c) The functor is representable. The representing ring is a λ-ring and is the free λ-ring on one generator. The relation (a)–(b) is a well known part of monad and comonad theory (from the early days of this theory). Every monad or comonad comes from an adjoint pair of functors, see [284, 33], and of course cofreeness of (A) −→ A translates into the observation that is right adjoint to the functor λ-Ring −→ CRing that forgets about the exterior product structure. Here is how the relation (b)–(c) goes.75 Let C be a category and let (T, μ, ε) be a comonad in C. Now let (Z, z ∈ T(Z)) represent the functor T . That is, there is a functorial bijection C(Z, A) −→ T(A), f → T(f )(z). The comonad structure gives in particular a morphism σ: Z −→ TZ, viz the image of idz under μz : T(Z) = C(Z, Z) −→ T(T(Z)) = C(Z, T(Z)). This deﬁnes a ‘coalgebra for T ’ structure on Z. Now let (A, σ) be a coalgebra for the comonad T and let a be an element of A. Consider the element σ(a) ∈ T(A) = C(Z, A). This gives a unique morphism of T -coalgebras that takes z into a. There are of course a number of things to verify both at this categorical level and to check that these categorical considerations ﬁt with the explicit constructions carried out in the previous subsections. This is straightforward. 16.76. Plethysm. (Very partial symmetric function formularium (4)). Given an element a ∈ A in a λ-ring A and a polynomial f ∈ Symm deﬁne αa (f ) = βf (a) = f λ1 a, λ2 a, · · · (16.77) where in the last part of (16.77) f is seen as a polynomial in the elementary symmetric functions; i.e. in the expression for f as a polynomial in the elementary symmetric functions these elementary symmetric functions are replaced by the exterior powers of the element a. If f is seen as a polynomial in the complete symmetric functions the same result is obtained by replacing these with the symmetric powers of the element a. This can lead to confusion so it is better to have a description that does not depend 75 I have not found this relation in the monad literature (which does not mean it is not there somewhere). There should be a general theorem to this effect also in a more general context than ‘categories of sets with structure.’

Witt vectors. Part 1

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on what free polynomial basis is used for Symm. That goes as follows. Give a ∈ A and f ∈ Symm Let αa : Symm −→ A be the unique morphism of λ-rings such that φa (h1 ) = φa |e1 | = a. Then αa (f ) = βf (a)

(16.78)

This looks at a ∈ A as something that deﬁnes (and is deﬁned by) a morphism of λ-rings from the free λ-ring on one generator to A. Another way to look at (16.77) is as the deﬁnition of a functorial operation on λ-rings deﬁned by the polynomial f . This is the most used way to look at (16.77), (16.78). In particular Symm itself is a λ-ring, and so, taking a g ∈ Symm for a in (16.77), (16.78) there results a new kind of composition of polynomials f ◦ g = αg (f ) = βf (g) = f λ1 g, λ2 g, · · · (16.79) This composition law on Symm is called plethysm. More precisely it is called ‘outer plethysm.’ There is also something called ‘inner plethysm’ which has to do with the λ-ring structures on the homogeneous summands Symm(n) of Symm.76 In the representation theoretic incarnation of Symm in terms of representations of the general linear groups, outer plethysm corresponds to composition of representations. In terms of the λ-ring interpretation mentioned above it also corresponds to composition: the functorial operation on λ-rings deﬁned by f ◦g on λ-rings is the composition of the functorial operations deﬁned by f and g: βf ◦g (a) = βf βg (a) (16.80) Plethysm is associative (f ◦ g) ◦ h = f ◦ (g ◦ h)

(16.81)

This is a special case of (16.80). 16.80. Calculation of plethysms. Write g ∈ Symm ⊂ Z as a sum of monomials in the ξ g = τ cτ ξ τ and write (formally)

c (1 + yi t) = 1 + ξτ t τ (16.81) τ

then f ◦ g = f (y1 , y2 , · · ·)

(16.82)

where f is seen as a symmetric polynominal in the ξ. 76 These homogeneous summands are rings under the second multiplication on Symm. In the representation theoretic incarnation of Symm they correspond with the ring of (virtual) representations R(Sn ) of the symmetric groups Sn . The exterior powers there are the exterior powers of representations.

426

M. Hazewinkel

Thus for instance f ◦ pn = f ξ1n , ξ2n , · · · , a plethysm that was used in 9.93 above when discussing the Witt symmetric functions which give the Witt coordinates of a power series. To see that (16.82)–(16.83) ﬁts with the deﬁnition, ﬁrst note that for the λ-ring structure on Z[ξ], λ1 (ξi ) = ξi , λn (ξi ) = 0 for n ≥ 2. Thus σt (ξi ) = (1 − ξi t)−1 and because σt is a morphism of rings

−c σt (g) = 1 − ξτ t τ (16.83) τ

and so λt (g) = σ−t (g)−1 =

1 + ξτ t

cτ

=

(1 + yi t)

τ

so that the elementary symmetric functions in the y are the λn g. 16.84. Distributivity of the functorial plethysm operations. The plethysm operations βf on λ-rings A are not additive. But they are distributive over addition and multiplication in a Hopf algebra way as follows. Let

μS (f ) = fS,i ⊗ fS,i , μP (f ) = fP,i ⊗ fP,i (16.85) i

Then βf (a + b) =

i

i

fS,i (a)fS,i , (b),

βf (ab) =

i

fP,i (a)fP,i (b)

(16.86)

This is seen as follows given a, b ∈ A, they deﬁne Witt vectors σt (a), σt (b) ∈ (A) which, in turn, are morphisms of rings Symm −→ A. Because σt is a morphism of rings the Witt vectors of a + b and ab are the Witt vector sum and Witt vector product ϕa ,ϕb of the Witt vectors of a and b. As morphisms of rings Symm −→ A. Witt vectors are summed and multiplied as follows. The sum Witt vector corresponds to the composite ϕa ⊗ϕb

μS

mA

Symm −→ Symm ⊗ Symm −−−→ A ⊗ A−→A and the product Witt vector corresponds to the composite ϕa ⊗ϕb

μP

mA

Symm −→ Symm ⊗ Symm −−−→ A ⊗ A−→A Formula (16.86) follows. For those polynomials for which there are nice formulas for the sum comultiplication morphism μS or for the product comultiplication μp (16.85) and (16.86) can give useful formulas for calculating plethysm operations. For instance, using 10.7 (sλ ◦ f ) (sκ/λ ◦ g) sκ ◦ (f + g) = (16.87) λ⊂κ

It also follows (again) from (16.85), (16.86) that the plethysm operations βpn on λ-rings deﬁned by the power sum symmetric functions pn are both additive and

Witt vectors. Part 1

427

multiplicative; they are of course the Adams operations (explaining once more why these are often called “power operations.’ Rather few of the plethysm operations deﬁned by a polynomial are additive: only those that come from a sum of power sum symmetric functions. On the other hand there are all the additive operations generated by the Frobenius, Verschiebung and homothety operations (the Cartier ring). It is unknown whether all these together with the plethysm operations generate all operations on the functor . There is quite a bit of literature on plethysm, mostly in a representation theoretic context and much is on how to calculate it in special cases. A sampling is [8, 66, 82, 282, 313, 394, 395], [312, 306, 43, 42, 429]. 17. Necklace rings There is (for suitable rings) a third coordinatization of the unital power series over a ring A, i.e. the elements of (A) = 1 + tA[[t]], besides the power series coordinates and Witt vector coordinates considered so far. These coordinates go by the name necklace coordinates. The three systems of coordinates are related by 1 + a1 t + a2 t + a2 t + · · · = 2

2

∞

1 − xi t

i=1

i −1

=

∞

1 − ti

−ci

(17.1)

i=1

Unlike in the case of Witt vector coordinates it is not always possible to ﬁnd necklace coordinates. It can certainly be done for the case that A is a Q-algebra, but there is but little interest in that as in that case W(A) ∼ = (A) ∼ = Gh(A) = AN But there are quite a few rings A that are not Q-algebras for which such a representation in necklace coordinates can always be found with the ci ∈ A; in particular the rather important case A = Z. This will be discussed a bit further below. On the other hand this can not be done for the case that A is a ring of polynomials, in particular not for A = Symm which is the context in which the universal example of a one power series lives; that is the power series h(t) = 1 + h1 t + h2 t 2 + h3 t 3 + · · ·

(17.2)

from which all others are obtained. Why I like to call the coordinates on the right hand side of (17.1) by the name necklace coordinates will become apparent in a minute. 17.3. Necklace polynomials. Consider a totally ordered alphabet of α letters, say {1, 2, · · · , α}. A word over this alphabet is primitive (also called aperiodic) if it is not a concatenation power of a strictly smaller word. A word is Lyndon if it is strictly smaller in the lexicographic order than any of its (non-identity) cyclic permutations. So in particular it is primitive. For the equivalence of this deﬁnition of Lyndon word with the one used in 16.71 see e.g. [273], Section 4.4.

428

M. Hazewinkel

A necklace (also called circular word) is an equivalence class under cyclic permutations of words. A necklace is primitive if the words in the equivalence class are primitive. Thus the Lyndon words can be regarded as a systematic choice of representatives of primitive necklaces. The formula for the number of primitive necklaces of length n on α letters, i.e. with α different colors of beads, has been known since Colonel Moreau, [298], in 1872. It is μ(d)αn/d (17.4) M(α; n) = n−1 d|n

These expressions, seen as polynomials in an indeterminate α are known as the necklace polynomials. Note that though these polynomials are integer valued for every integer argument they do not have integer coefﬁcients. For instance for a prime number p M(α; p) = p−1(αp − α)

(17.5) The same expression (17.4) turns up in other contexts. For instance in the theory of free Lie algebras. Consider the free Lie algebra generated by α symbols. Give each symbol weight one. The free Lie algebra (over the integers or any ﬁeld) is then graded and the graded part of weight n has rank M(α, n). This can be seen for instance by the so called Hall set construction of a basis (as an Abelian group) for the free Lie algebra, using the set of Lyndon words as a Hall set. See Chapter 5 in [334] for details. In this context of free Lie algebras (17.4) is known as the Witt formula, [418]. In loc. cit., p. 153, Witt writes: “Es ist merkwürdig, dass diese Rangformel übereinstimmt mit der bekannten Gausschen Formel für die Anzahl der Primpolynome xn + a1 xn−1 + · · · + an im Galoisfeld von q Elementen.” Later, Solomon Golomb, [174], found indeed a bijection between primitive necklaces and irreducible polynomials. This correspondence is not yet entirely satisfactory in that it depends on the choice of a primitive element for the Galois ﬁeld. See also [335] for a discussion of this correspondence and other occurences of expression (17.4). Consider the power series in Witt coordinates

−1 1 − M(α, n)t n βn t n = (17.6) n≥0

n≥1

Then βn is the rank of the homogeneous component of degree n of the free associative algebra over Z in α symbols. This results from the Witt formula by using the Poincaré-Birkhoff-Witt theorem combined with the Milnor-Moore type result that says that the free associative algebra is the universal enveloping algebra of the free Lie algebra on the same set of symbols. For some further manifestations of the necklace polynomials and related expressions in such varied ﬁelds of inquiry as the Feynman identity, the elliptic modular function, multiple zeta values, (symbolic) dynamics and ﬁxed points, formal groups, see also [70, 101, 113, 218, 234, 261, 260, 292, 299, 316, 317]. 17.7. Necklace polynomial formulas. Here are two interesting formulas from [291]. M(αβ; n) = (i, j)M(α; i)M(β; j) (17.8) [i,j]=n

Witt vectors. Part 1

429

where (i, j) denotes the greatest common divisor of i and j, and [i, j] denotes their least common multiple. j M(β; j) (17.9) M(βr ; n) = n where the sum ranges over all j such that [j, r] = nr (which implies that j is a multiple of n). 17.10. Cyclotomic identity. The so called cyclotomic identity is M(α;n) ∞

1 1 = (1 − αt) 1 − tn

(17.11)

n=1

(It is a tradition to write it precisely this way.) All three identities (17.8), (17.8), (17.11) are identities about polynomials. As such it sufﬁces to prove them for integer values of the variables α, β which is (usually) done by combinatorial means. Cf e.g. [290, 291]. There is a very elegant ‘symmetric’ generalization of the cyclotomic identity due to Volker Strehl, [375] M(β;n) M(α;n) ∞ ∞

1 1 = (17.12)77 1 − αt n 1 − βt n n=1

n=1

There are some other generalizations which will be described later in a section devoted to generalized Witt vectors. 17.13. Motivational remarks regarding the necklace algebra functor. Now consider power series expressed by means of necklace coordinates as in (the right hand side of ) 17.1. Multiplying two such expression means adding their necklace coordinates. The multiplication deﬁned on the Witt vector ring is determined by 1 1 1 ∗ = (1 − αt) (1 − βt) (1 − αβt)

(17.14)

Taking into account distributivity, the cyclotomic identity (17.11) and formula (17.9) this dictates the following deﬁnition of the necklace algebra (and its ghost component morphisms). 17.15. Deﬁnition of the necklace algebra functor. As an Abelian group the necklace ring over a ring A is the inﬁnite product AN of all sequences (a1 , a2 , a3 , . . .), ai ∈ A with component wise addition. Two such sequences are multiplied according to the rule (a1 , a2 , a3 , · · · ) ∗ (b1 , b2 , b3 , · · · ) = (c1 , c2 , c3 , · · · ) cn = (i, j)ai bj

(17.16)

[i,j]=n

77 This has the ﬂavour of a ‘reciprocity formula’ and one wonders if it relates to some other ‘reciprocity results’ in mathematics.

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This is clearly functorial and this functor will be denoted Nr : CRing −→ CRing. There is a functorial ghost components ring morphism u : Nr(A) −→ Gh(A) = AN to the direct product of N copies of the ring A deﬁned by u(a) = (u1 (a), u2 (a), u3 (a), · · · ), un (a) = dad (17.17) d|n

Given a power series for which necklace coordinates exist, so that (17.1) holds, one then has for these coordinates w(x) = s(a) = u(c)

(17.18)

in the ghost ring Gh(A) = AN (with component-wise addition and multiplication). Necklace rings (in the sense deﬁned above78 are a special kind of convolution rings as deﬁned in [403, 402, 405]. For some ﬁrst investigations in the algebraic theory of necklace rings see [404, 322]. 17.19. Binomial rings. Obviously, from (17.1), necklace coordinates are hardly compatible with the presence of torsion. Further as x(x − 1) 2n x(x − 1)(x − 2) 3n (1 − t n )x = 1 − xt n + t − t + · · · (17.20) 2! 3! it seems necessary for necklace coordinates to exist over a ring A to require that together with an x ∈ A it also contains the binomial coefﬁcients (n!)−1 x(x−1) · · · (x− n + 1). This leads to the idea of a binomial ring. A (commutative unital) ring A is said to be binomial if it is torsion free (as a Z-module) and if together with any x in it it also contains the binomial coefﬁcients x(x − 1)(x − 2) · · · (x − n + 1) x = (17.21) n n! These rings have made their appearance before in 1958, in the fundamental work of Philip Hall on the theory of nilpotent groups, [186], often referred to as the ‘Edmonton notes.” Obviously any Q-algebra is binomial. But there are many rings that are binomial that are not algebras over the rationals. First and foremost the ring of integers Z and certain rings of functions with values in the integers (under pointwise addition and multiplication), notably polynomials. 17.22. Free binomial rings. Let X = {Xi : i ∈ I} be a set (of indeterminates). By deﬁnition IVal[X] is the ring of all polynomials with rational coefﬁcients that take integer values on integers; i.e. integer valued polynomials. These are also sometimes called numerical polynomials. 78 There is another kind of object in this more or less same corner of algebra that has ‘necklace’ in its name. Viz necklace Lie algebras, necklace Hopf algebra, necklace Lie coalgebra, see [55, 151, 170, 352]. These have little to do with the necklace approach to Witt vectors.

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Examples are the binomial coefﬁcients polynomials Xi (Xi − 1)(Xi − 2) · · · (Xi − n + 1) Xi = n n!

(17.22)

and it is a theorem that the monomials in these form a basis of IVal[X] over the integers (as an Abelian group), see [45], §45, p. 240ff. These are indeed the free binomial rings in the technical sense that the functor IVal : Set −→ BinRing (where BinRing stands for the sub category of binomial rings in CRing) is left adjoint to the forgetful functor the other way, see [134], p. 168, proposition 2: BinRing(IVal[X], A) ∼ = Set(X, A)

(17.23)

A monograph on integer valued polynomials is [80].79 Other examples of binomial rings are the p-adic integers, the proﬁnite completion ˆ of the integers and localizations of the integers, and a special kind of λ-rings. Z 17.24. Binomial rings vs λ-rings. Let A be a torsion free ring such that xp ≡ x mod p for all prime numbers p. Then, taking ϕn = id for all n the conditions of the Wilkerson theorem 16.42 are satisﬁed and so there is a λ-ring structure on A for which all the Adams operations are the identity. For this λ-ring structure x n , λt (x) = (1 + t)x , σt (x) = (1 − t)−x (17.25) λ (x) = n and thus A is a binomial ring. To see (17.25) simply calculate the determinant of the matrix Mx = (mi,j ), mi,j = x for i ≥ j, mi,i+1 = i, mi,j = 0 for j ≥ i + 2, see (16.41). Actually one can do a little better and prove that a λ-ring for which all the Adams operations are equal to the identity is automatically torsion free, see [134], section 5. Inversely let A be a binomial ring then taking the (tentative) exterior product operations to be the binomial coefﬁcients one gets a ring with exterior products with associated Adams operations that are all the identity and hence a λ-ring structure. 17.26. Frobenius and Verschiebung on necklace rings. From the above it is clear that for binomial rings necklace coordinates always exist and that the necklace ring over a binomial ring is isomorphic (functorially) to the ring of Witt vectors over that ring. Using this isomorphism one can of course transfer the Frobenius and Verschiebung operations to necklace rings. Actually these are always deﬁned. Here are the explicit formulas. For c = (c1 , c2 , c3 , · · · ) ∈ Nr(A), fr c = (b1 , b2 , b3 , · · · ) n−1 jcj Vr c = (0, 0, · · · , 0, c1 , 0, 0, · · · , 0, c2 , · · · ) (17.27) bn = [j,r]=nr

r

79 As far as I know this is also the only monograph on the topic.

r

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and one easily proves the usual formulas; either directly or via the ghost components or via …, such as Vr Vs = Vrs , fr fs = frs , fr Vr = [r], fr Vs = Vs fr if (r, s) = 1

(17.28)

17.29. Witt vectors vs necklaces in general. Let A be a torsion free ring.80 Consider the mapping ψA : Nr(A) −→ (A), (c1 , c2 , c3 , · · · ) →

∞

(1 − t n )−cn

(17.30)

n=1

This is by the very deﬁnition of addition and multiplication on Nr(A) a morphism of rings and it is compatible with the Frobenius and Verschiebung morphims on the two sides. When A is a binomial ring it is an isomorphism. When A is not binomial this is not the case. However, it is an isomorphism for AQ = A⊗Q and it is (of course) dead easy to describe the subring of Nr(AQ ) which corresponds to the subring (A) ⊂ (AQ ). For each α ∈ A let M(α) be the necklace vector M(α) = (M(α; 1), M(α; 2), M(α; 3), · · · ) ∈ Nr(AQ )

(17.31)

Then by the cyclotomic identity and the deﬁnition of the Verschiebung operations ψAQ (M(α)) = (1 − αt)−1 ,

ψAQ (Vr M(α)) = (1 − αt r )−1

and it follows that ψAQ induces an isomorphism ∞ # ψAQ Vr M(αr ) : αi ∈ A −−→ (A) ⊂ (AQ )

(17.32)

(17.33)

r=1

By the left half of (17.32) it is clear that the necklace vectors M(α) are the analogs of Teichmüller representatives. For a binomial ring, like the integers, it follows from (17.31)–(17.33) that each necklace vector can be written uniquely as an inﬁnite sum ∞

Vr M(αr ), M(αr ) = (M(αr ; 1), M(αr ; 2), M(αr ; 3), · · · ) ∈ Nr(A) (17.34)

r=1

17.35. Modiﬁed necklace rings. The description (17.33) of the ring of Witt vectors in terms of necklace vectors is not very satisfactory or elegant. One can in fact do a 80 As far as I know no investigations have been carried out on the necklace ring over A when A is

not torsion free, for instance when A is a ﬁnite ﬁeld. Yet it is in that setting that something interesting may happen that is different from what happens in the case of the Witt vectors. Also the analogue of the p-typiﬁcation endomorphism deﬁned by the Frobenius and Verschiebung operations has not been investigated for necklace rings.

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good deal better using modiﬁed necklace polynomials, [316]. In loc. cit. Young-Tak Oh deﬁnes modiﬁed necklace polynomials over a λ-ring A which explicitly involve the Adams operations as follows M(r; n) = n−1 μ(d )d (r n/d ) (17.36) d|n

He also introduces much related81 multivariable versions of the necklace polynomials M(X; n) = n−1 μ(d )pd (X) (17.37) d|n

where the pd ’s are the power sums in the X’s. These polynomials satisfy very similar properties to (17.7) and (17.8) and lead to a deﬁnition of necklace rings for λ-rings which are isomorphic to the rings of Witt vectors. It would be very nice if there were combinatorial interpretations of these modiﬁed necklace polynomials. 17.38. Adjoints of the inclusion BinRing ⊂ CRing. Now consider the inclusion of the binomial rings into the category of rings. This inclusion functor has both a left adjoint (free objects), BinU , and a right adjoint (cofree objects), BinU , characterized and deﬁned by the functorial properties BinRing(BinU (A), B) ∼ = CRing(A, B), BinRing(B, BinU (A)) ∼ = CRing(B, A) for binomial rings B and rings A. The construction of BinU (A) is much like IVal[X] compared to Z[X]. For BinU (A) take (A) and take the subring of all elements on which all the Frobenius operations are the identity. For more details and proofs see [134]. 17.39. The functors BinW. The sum comultiplication μs and the product comultiplication μp on Z[h] that deﬁne the ring valued functor of the big Witt vectors, extend uniquely to IVal[h1 , h2 , h3 , · · · ] and IVal[h1 , h2 , h3 , · · · , hn ] to deﬁne coring objects and hence ring-valued sub functors of the Witt vectors and the truncated Witt vectors. These would seem to merit some investigation. Even the simplest one IVal[h1 ] already has thought provoking properties, see [45], p. 241ff. 17.40. Carryless Witt vectors. Like the real numbers in decimal notation the Witt vectors, when added or multiplied, involve ‘carry-overs.’ The necklace approach to them can be used to describe a carry-less version, [291]. The price, however, is high, too high in my opinion, in that Witt vectors in this approach are seen as equivalence classes rather than single objects.

81 This again involves the splitting principle technique/philosophy.

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17.41. To ponder and muse about. A nice theory of necklace rings as in [316] works precisely for λ-rings, i.e. the very objects which are coalgebras for the main functor involved, the one of the Witt vectors. There is something circular about this; almost incestuous, of much the same ﬂavour as one often meets in various parts of category theory. To me this is something that needs to be pondered and mulled over a bit. 17.42. Apology. Within the published literature on necklace algebras one ﬁnds the notion of what are called aperiodic rings. The formulas involved are, in my opinion, essentially empty given the necklace formulas and, better, Witt vector formulas. So in this chapter 1 will not discuss these aperiodic rings. 17.43. A selection of references on necklaces and necklace algebras is [21, 22, 55, 70, 115, 120, 144, 145, 174, 260, 261, 291, 294, 303, 314, 315, 316, 322, 329, 399, 401, 404]. 18. Symm vs ⊕ R(Sn ) n

That symmetric functions and representations of the symmetric groups representations of the general linear groups have much to do with one another has been known since the early days of the previous century (Alfred Young, Issai Schur, Georg F Frobenius, . . .). The realization that these things become more elegant and better understandable from the Hopf algebra point of view is of a more recent date, [269, 268, 239, 425]. Here, mostly, the latter approach will be outlined. For more details see [199], Chapter 4. First, however, here is a streamlined version of what might be called classical Schur-Frobenius theory as presented in [281], pages 112–114. 18.1. The ring R(S). Let Sn be the group of permutations on n letters, usually taken to be {1, 2, . . . , n}. For each n let R(Sn ) be the free Abelian (Grothendieck) group spanned by the irreducible representations of the symmetric group Sn ; or, what is the same, the Grothendieck group of isomorphism classes of (complex) representations of Sn , or, what is again the same, the free Abelian group with a basis the irreducible characters of Sn . An element from R(Sn ) that comes from an actual representation will be called real (as opposed to virtual (not as opposed to complex)). These are the ones that are nonnegative integral sums of irreducible representation. Other elements of R(Sn ) are sometimes referred to as virtual representations. As an Abelian group R(S) is the direct sum of all these groups of representations: ∞

R(S) = ⊕ R(Sn ) n=0

(18.2)

where, by decree, R(S0 ) = Z. A product is deﬁned on R(S) as follows. Take a representation ρ of Sp and a representation σ of Sq . Taking the (outer) tensor product gives a representation ρ ⊗ σ of Sp × Sq on the tensor product of the representation spaces of ρ and σ. Now consider Sp × Sq as the subgroup of Sn = Sp+q of those permutations that take the set of the ﬁrst p elements into itself and that take the set

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of the last q elements into itself. Now induce the representation ρ ⊗ σ up to Sn . Further take 1 ∈ Z = R(S0 ) to be the unit element. This deﬁnes an associative and commutative multiplication on R(S) that makes it into a graded ring. As a rule this outer product of two irreducible representations is not irreducible and determining the multiplicities of the irreducible representations that occur in it is and always has been a major part of the representation theory of the symmetric groups (LittlewoodRichardson rule82 ). 18.3. Scalar product.83 If f , g are functions on a ﬁnite group G their scalar product is deﬁned by 1 f(x)g(x−1 ) (18.4) f, gG = #G x∈G

This scalar product is used to deﬁne a scalar product on all of R(S) by fn , gn Sn f, g = where f =

n≥0 ˙

fn , g =

(18.5)

gn ∈ R(S) (and 1, 1 = 1 for 1 ∈ R0 (S) = Z).

18.6. Characteristic map. Let w ∈ Sn be a permutation on n letters. It decomposes as a product of disjoint cycles and the lengths of these cycle deﬁne a partition λ(w) of n called the cycles type of w. Deﬁne a mapping ψ : Sn −→ Symmn ,

w → pλ(w)

(18.7)

where pλ(w) is the power sum monomial, see (9.61), deﬁned by the cycle type of w. The next step is the deﬁnition of a morphism of Abelian groups called the characteristic map ch : R(S ) −→ SymmC = Symm ⊗Z C

(18.8)

as follows. If f is a (virtual) character of Sn 1 ch(f ) = f, ψSn = f(w)ψ(w−1 ) n! w∈Sn

=

1 f(w)ψ(w) = z−1 λ fλ pλ n! w∈Sn

(18.9)84

wt(λ)=n

where fλ is the value of f on the cycle class λ and zλ is the number from (9.62) that gives the scalar product of the power sum monomials with themselves and where it 82 [230], theorem 2.8.13, p. 93. 83 This inner product is sometimes called ‘Hall inner product.’ 84 Where for the second expression in (18.9) deﬁnition (18.4) is extended a bit in that g has its values in

Symm.

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M. Hazewinkel

is used that the cycle class of w ∈ Sn is the same as that of w−1 . (Recall that f as a character has the same value on each element of a conjugacy class). It follows that, using also pλ , pμ = zλ δλ,μ , z−1 (18.10) ch(f ), ch(g) = λ fλ gλ = f, gSn wt(λ)=n

and hence that ch is an isometry. The basic theorem is now that 18.11. Theorem. The characteristic map is (induces) an isometric isomorphism from R(S) onto Symm. There are a certain number of things to prove. But this is not a text on representation theory, let alone a text on the representation theory of the symmetric groups, that vast and fascinating subject; so I will restrict myself to a brief sketch. First, ch is multiplicative. This is handled by Frobenius reciprocity (which will also turn up later in the Hopf algebra approach). Next one shows that the identity character ηn , i.e. the character of the trivial representation of Sn corresponds under ch to the complete symmetric function hn . (The sign character corresponds to the elementary symmetric function en .) Further for each partition λ of n deﬁne (18.12) χλ = det ηλi −i+j 1≤i,j≤n ∈ R(Sn ) These are (possibly virtual) characters of Sn . Their images under ch are the Schur functions and so χκ , χλ = δκ,λ and as ch is an isometry they are up to sign irreducible characters. As the number of conjugacy classes of Sn is equal to the number of partitions of n they must form a basis for R(Sn ) and so ch indeed induces an isomorphism of R(Sn ) onto Symm. It remains to verify that the χλ are in fact real characters (as opposed to virtual) which is done by checking their value on the trivial permutations. 18.13. Enter Hopf algebras. The topologist knows Symm as H ∗ (BU; Z), the cohomology of the classifying space BU of the unitary group. And he remembers that this is a Hopf algebra. So wouldn’t it be nice if R(S ) were a Hopf algebra too and if ch were an isomorphism of Hopf algebras. This is indeed the case. The ﬁrst to notice that R(S) is a Hopf algebra would appear to have been Burroughs, [79]. It was, however, Arunas Liulevicius, [269, 268], who ﬁrst systematically exploited this point of view. To quote a bit more from the second of his two papers: “The aim of this paper is to present a ridiculously simple proof of a theorem on representations rings of the symmetric groups which concisely presents theorems of Frobenius, Atiyah, and Knutson” (and Schur in my view of things). Whether this approach is really simpler than the very streamlined presentation of Macdonald outlined above is debatable. A Hopf algebra is a heavy structure. But it certainly brings in new and more functorial and universal points of view, which are, I believe, important.

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18.14. The Hopf algebra structure on R(S). For a composition α = [a1 , a2 , · · · , am ] of weight n the corresponding Young subgroup is Sα = Sa1 × Sa2 × · · · × Sam . It consists of all permutations that map all the sets of letters {1, · · · a1 }, {a1 +1, · · · , a1 + a2 }, · · · {a1 +· · · am−1 +1, · · · , a1 +· · ·+am = n} to themselves. Deﬁne a coproduct structure by μS (ρ) = 1 ⊗ ρ +

n−1 i=1

n

ResSSni ×Sn−i (ρ) + ρ ⊗ 1 ∈ ⊕ R(Si ) ⊗ R(Sn−i ) i=0

(18.15)

and a counit (augmentation) εS : RS −→ Z, ε =

id

on R(S0 ) = Z

0

on R(Sn ), n ≥ 1

(18.16)

The basic fact is now that together with the ring structure already deﬁned and used this makes R(S) a Hopf algebra. The harder part of this is to prove the compatibility of the multiplication with the comultiplication. This is taken care of by the Mackey double coset theorem which describes what happens when a representation is ﬁrst induced up from a subgroup and then restricted to another subgroup, see [199], Chapt. 4 for details. Actually, it is the projection formula which does the job (together with a description of double cosets of Young subgroups of the type Si × Sn−i ⊂ Sn . This ‘induction restriction projection formula’ is the following:85 18.17. Theorem. Projection formula, Frobenius axiom. Let G be a ﬁnite group with subgroup H, and let V be an H-module and W a G-module, then G G Ind G (18.18) H V ⊗ ResH (W ) = Ind H (V ) ⊗ W 18.19. Hopf algebras continue their insidious work. For an account of how to use the Hopf algebraic structure so far described in the representation theory of the symmetric groups, see the two papers of Liulevicius already quoted. Here I will now continue to describe some more Hopf algebraic structure, culminating in the Zelevinsky structure theorem. This involves further fairly heavy machinery, and certainly does not give the easiest way to get at the representation theory of the symmetric groups. But this way is, in my view, quite important. As to the various bits of structure on R(S ) the situation is now as follows: (i) R(S ) is a connected graded Z module. (ii) There is a preferred basis consisting of 1 and the irreducible representations with a corresponding inner product for which this basis is orthonormal. (iii) Multiplication is positive (because taking the outer tensor product of true (real) representations and than inducing up gives a true representation). 85 Caveat: this is not the same instance of a projection formula as occurred i theorem 13.48.

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(iv) Comultiplication is positive (because restricting a true representation yields true representations. (v) Comultiplication is multiplication preserving (or, equivalently, the multiplication is comultiplication preserving). This is an immediate consequence of the Mackey double coset theorem combined with the description of double cosets of Young subgroups. (vi) Comultiplication and multiplication are dual to each other with respect to the inner product. This follows from Frobenius reciprocity in the form that induction is adjoint to restriction (both on the left and on the right) combined with the fact that the scalar product of two characters on a group counts the number of irreducible representations that they have in common. (vii) The counit morphism is a morphism of algebras. (viii) There is just one element of the preferred basis that is primitive. This is the unique irreducible representation of S1 . Indeed for all n ≥ 2 a real representation of Sn must restrict to some real (as opposed to virtual) representation of S1 × Sn−1 . 18.20. PSH algebras. The acronym ‘PSH’ stands for ‘positive selfadjoint Hopf.’ This implies sort of that there is also a positive deﬁnite inner product and that we are working over something like the integers or the reals, where positive makes sense. What is not mentioned is that these Hopf algebras are also supposed to be connected and graded. As will be seen the assumptions ‘positive, selfadjoint, graded’ are all three very strong; so strong that these algebras can actually be classiﬁed. Indeed they are all products of one example86 and that example is the Hopf algebra of the symmetric functions. The notion is due to Zelevinsky, [425], and the classiﬁcation theorem is his. The precise deﬁnition is as follows. A PSH algebra is a connected graded Hopf algebra over the integers, so that H = ⊕ Hn , n

H0 = Z,

rk(Hn ) < ∞

(18.21)

which is free as an Abelian group and which comes with a given, ‘preferred’ homogeneous basis {ωi : i ∈ I} = B. Deﬁne an inner product , on H by declaring this basis to be orthonormal. Then the further requirements are: Selfadjointness: xy, z = x ⊗ y, μ(z), ε(x), a = x, e(a), x, y, z ∈ H, a ∈ Z r Positivity: let ωi ωj = ai,j ωr , μ(ωr ) = bri,j ωi ⊗ ωj , r

then

(18.22)

i,j

r , bri,j ai,j

≥0

(18.23)

86 More precisely they are all products of that one example, but the factors are possibly degree shifted.

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The Zelevinsky classiﬁcation theorem now says that a PSH algebra with just one primitive among the preferred basis elements is isomorphic to the Hopf algebra of the symmetric functions (as a Hopf algebra). The proof proceeds by the inductive construction in any PSH algebra with just one primitive preferred basis element p of a series of elements that behave just like the hn , en , pn of symmetric function theory. The key observation here is that the powers of p must involve all preferred basis elements. In the representations theoretic case R(S ) this corresponds to the fact that by deﬁnition of the outer multiplication pn is the regular representation of Sn . Here p is the identity representation of S1 . For the details of the proof see [425], [199] Chapter 4.87 18.24. Bernstein morphism.88 The ﬁnal step of the proof involves a construction that I think of particular interest, that is not yet well understood, and that deserves more study. Let H be any graded commutative and associative Hopf algebra. Let μn (x) = xi,1 ⊗ xi,2 ⊗ · · · ⊗ xi,n

(18.25)

i

be the n-fold comultiplication written as a sum of tensor products of homogeneous components. Now deﬁne βn : H −→ H[ξ1 , ξ2 , . . .] = H ⊗ Z[ξ1 , ξ2 , . . .] by βn (x) =

deg(xi,1 ) deg(xi,2 ) deg(x ) ξ2 · · · ξn i,n

xi,1 xi,2 · · · xi,n ξ1

(18.26)

i

Because H is coassociative and cocommutative this is symmetric in the variables ξ1 , · · · , ξn so that there is an induced algebra morphism89 βn : H −→ H ⊗ Z[h1 , . . . , hn ] and because H is graded this stabilizes in n giving the Bernstein morphism90 β : H −→ H ⊗ Symm

(18.27)

In the case that H is Symm this turns out to be the product comultiplication morphism!. The ﬁnal step of the proof of the Zelevinsky classiﬁcation theorem is now to 87 The proof as written down in loc. cit. is just my own attempt to write down Zelevinsky’s proof in a way that I could understand it. 88 The construction is due to Joseph N Bernstein. 89 The fact that this is a morphism of algebras uses commutativity; otherwise multiplication is not an algebra morphism. 90 For a general graded commutative Hopf algebra the Bernstein morphism deﬁnes a coaction of Symm on H and then by duality also an action of Symm on the graded dual of H.

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compose the Bernstein morphism with a morphism H −→ Z that takes the inductively constructed analogues of the hn all to one. In the case H = Symm this is the counit morphism of the product comultiplication morphism μP .91 As promised in 11.39 above, there now follow a few remarks on (0, 1)-matrices and their links with Witt vectors and representation theory. This requires some preparation. 18.28. Majorization ordering. Let α = (a1 , a2 , · · · , an ) and β = (b1 , b2 , · · · , bn ) be two vectors of non-negative real numbers of the same l1 -norm, i.e. a1 + a2 + · · · + an = b1 +b2 +· · ·+bn . Denote by α = (a1 , a2 , · · · , an ) a reordering (rearrangement, permutation) of α such that a1 ≥ a2 ≥ · · · ≥ an . The majorization ordering is now deﬁned by α ≥maj β ⇔

r

ai ≥

i=1

r

bi ,

r = 1, 2, · · · , n

(18.29)

i=1

This ordering occurs is many parts of mathematics under many different names: majority ordering, dominance ordering, natural ordering, specialization ordering, Snapper ordering, Ehresmann ordering, mixing ordering. Parts of mathematics where it plays an important role include: families of algebraic geometric vectorbundles, families of representations, families of nilpotent matrices, Grassmann manifolds, control theory, representation theory, thermodynamics, convex function theory (Schur convex functions92 ), doubly stochastic matrices, (0, 1) matrices, inequality theory (Muirhead inequalities, a far reaching generalization of the geometric mean-arithmetic mean inequality), representation theory, . . .. See [3]. Many of these uses of the majorization ordering are strongly related, see [202]. 18.30. Conjugate partition.$ Let α = [a1 , a2%, · · · , am ] be a partition of n. Then the

conjugate partition αconj = a1 , a2 , · · · , am

is deﬁned by ai = #{j : aj ≥ i}

(18.31)

[4, 4, 3, 2, 1, 1, 1]conj

So, for instance, = [7, 4, 3, 2]. (If the partition is displayed as a diagram (either in the French or Anglo-Saxon manner), the conjugate partition looks at columns instead of rows.) There are (at least) three93 applications of the majorization ordering in the representation theory of the symmetric groups. In addition there is the Gale-Ryser theorem which is of immediate relvance here as it deals with the existence of (0, 1)-matrices 91 Note that without the characteristic map isomorphism or the Zelevinsky theorem itself, it is not yet

clear that Symm is PSH (but R(S) is). The trouble is positivity; speciﬁcally the fact that the product of two Schur symmetric functions is a nonnegative linear combination of Schur functions. This seems a fact that is not so easy to establish directly (without going to representation theory) (and, hence, is a bit of a blemish on symmetric function theory). 92 Same Schur; totally different topic. 93 Another (related one), due to Kraft and de Concini, is too far from the present topic to discuss.

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and thus has things to say about the second comultiplication on Symm (and hence the multiplication of Witt vectors. 18.32. Gale-Ryser theorem. Let α and β be two partitions then there is a (0, 1)−matrix with row sum α and column sum β if and only if αconj ≥maj β. Of course a similar theorem holds for compositions instead of partitions; this amounts to taking permutations of columns and permutations of rows. For example take the (0, 1)−matrix ⎛ ⎞ 0 1 0 0 1 M = ⎝1 0 1 0 1⎠ 0 1 0 1 0 This one has as row sum the composition [2, 3, 2] with associated partition [3, 2, 2]; and the column sum is the composition [1, 2, 1, 1, 2] with associated partition [2, 2, 1, 1, 1]. Also [3, 2, 2]conj = [3, 3, 1]. And, indeed [3, 3, 1] ≥maj [2, 2, 1, 1, 1]. Of the three theorems in the representation theory of the symmetric groups that involve the majorization ordering the one closest to the present concerns (Witt vectors) is the Snapper Liebler-Vitale Lam theorem. 18.33. Snapper Liebler-Vitale Lam theorem. Let α and β be partitions and let Sα be the Young subgroup deﬁned by α. Then ρβ , the irreducible representation corresponding to β, occurs in Ind SSnα (I) if and only if α ≤maj β. Here I stands for the trivial representation of Sα . For a proof see [203] and/or the references therein. For completeness sake and possible future applications, below there is the RuchSchönhofer theorem. Under the isomorphism between Symm and R(S) the trivial representation of Sn corresponds to the complete symmetric function hn and the sign representation ASn corresponds to the elementary symmetric function en . So this theorem could certainly be relevant. 18.34. Ruch-Schönhofer theorem. The representations Ind SSnα ISα and Ind SSnβ ASβ have an irreducible representation in common (which is the same as saying that their inner product is nonzero), if and only if α ≤maj β. 18.35. Outer plethysm and inner plethysm. In section 16.67 the composition operator ‘outer plethysm’ on Symm made its appearance. But Symm is isomorphic to R(S ) as Hopf algebras (with extra product multiplication and extra coproduct multiplication with counit). Thus there is an outer plethysm operation on R(S). The problem is to describe this operation in representation theoretic terms. This has been called the ‘outer plethysm problem.’ ∞

The other way: each of the summands of R(S ) = ⊕ R(Sn ) is a λ−ring in its own n=0

right, giving rise to plethym operations called inner plethysm, and the ‘inner plethysm problem’ is to describe these in symmetric function terms when transferred to Symm.

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18.36. Outer plethysm problem. Outer plethysm gives in particular a composition of ISn , the trivial representation of Sn with any σ ∈ R(Sk ). The result is denoted hn (α) in [239], p. 135. The trivial (or identity) representation of Sn corresponds to hn ∈ Symm, hence the notation. In loc. cit., p. 135 Donald Ivar Knutson guesses94 the following representation theoretic description of hn (α). Given a representation α of Sk on V ﬁrst construct the induced representation of the wreath product Sn [Sk ] on V ⊗n , then induce up the representation obtained to Snk using the natural inclusion Sn [Sk ] ⊂ Snk . This ‘guess’ has since turned out to be correct, see [221, 313, 394]. 18.37. Inner plethysm problem. The λ−ring structure on R(Sn ) gives of course rise to Frobenius operators which via the isomorphism give rise to (degree preserving) Adams-Frobenius type operators on the homogeneous component of weight (degree) n of Symm. Using the inner product there are corresponding adjoint operators. These are described nicely in section 3 of [349]. Besides the ﬁve references just quoted here is a selection of further references on plethysm: [8, 43, 42, 59, 66, 82, 267, 265, 266, 282, 306, 312, 313, 341, 351, 385, 394, 395, 429]. 19. Burnside rings Let G be a ﬁnite group. The Burnside ring B(G) is one of the fundamental (representation like) rings attached to G. It is the Grothendieck ring of ﬁnite G-sets with sum and product inducted by disjoint union and Cartesian product respectively. Or, equivalently, the ring of permutation representations. So as a group B(G) is a ﬁnitely generated Abelian group with as (canonical) basis the transitive G-sets (orbits). According to some the Burnside ring was introduced by Andreas Dress in [117], where he proved a somewhat unexpected kind of result, viz. that a ﬁnite group is solvable if and only if the spectrum of its Burnside ring is Zariski connected, i.e. if and only if 0 and 1 are the only idempotents in B(G). Others attribute the introduction of the concept to Louis Solomon [372]. By now there is a substantial literature on Burnside rings: the ZMath database lists at the moment95 305 publications with “Burnside ring” in title or abstract. For a fair selection see the bibliography of [61]. “[The Burnside ring] is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed an analogue of the ring Z of integers for this category.” “The ring B(G) is also functorial with respect to G and subgroups of G and this leads to the Mackey functor or Green functor point of view. . . . The Burnside Mackey functor is a typical example of projective Mackey functor. It is also a universal object in the category of Green functors.” 94 His word. 95 19 April 2008.

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These are two quotes from the introduction of [61]. So if the Burnside ring of an arbitrary ﬁnite group is already as nice as all that, what about the Burnside ring of a really nice group like the integers. That one must then be super-nice. And it is. It turns out to be the ring of Witt vectors of the integers, W(Z), a discovery due to Andreas Dress and Christian Siebeneicher, [115, 120].96 19.1. G-sets. Let G be a group. A G-set is a set X together with an action (on the left) of G on it; that is a mapping G × X −→ X, (g, x) → gx such that g(h(x)) = (gh)(x), g, h ∈ G, x ∈ X and e(x) = x where e is the identity element of G. A morphism of G-sets is a mapping f : X −→ Y such that f(gx) = g(f(x)). This deﬁnes the category G-Set of G-sets. For x ∈ X the subgroup Gx = {g ∈ G : gx = x}

(19.2)

is the stabilizer of x ∈ X and Gx = {gx : g ∈ G}

(19.3)

is the orbit of x (or through x). If H is a subgroup multiplication on the left induces a G-set structure on the set of left cosets G/H which has a single orbit (a transitive G-set). The orbit Gx of an element is isomorphic as a G-set to G/Gx . A G-set X gives rise to a permutation representation ρ(X) over any ring of which the underlying module is the free module with basis X and with the action of G given by the the permutations of basis elements determined by the G-set structure of X. 19.4. Induction and restriction for G-sets. Let H be a subgroup of a group G. Then if X is a G-set retricting the action to H gives an H-set, deﬁning a functor ResG H : G − Set −→ H − Set. As is to be expected there is also a functor the other way called induction. Let Y be an H-set. Consider the set G × Y with the equivalence relation determined by (g, y) ∼ (gh−1 , hy), g ∈ G, h ∈ H, y ∈ Y

(19.5)

set of The equivalence classes is suggestively denoted G×H Y . (Note that (g, y) → gh−1 , hy deﬁnes an action of H on G × Y and that classes are the the equivalence orbits for this action.) Multiplication on the left g , (g, y) → (g g, y) induces an action of G on G ×H Y giving a G-set denoted Ind G H (Y ). It is obvious how to make : H − Set −→ G − Set. Induction is left adjoint to restriction this into a functor Ind G H (but not right adjoint in general). There is also a product formula (projection formula, Frobenius identity) X × Ind G (Y ) ∼ (19.6) = Ind G ResG (X) × Y H

H

H

For more details on this and some more related material see e.g. [409] p. 811, [61] pp. 744–745. 96 This also of course, given the ‘universality remark’just quoted, provides a seventh universality property of the Witt vectros. Working out what this one really means is still an open matter.

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Restriction and induction are compatible with the notions of the same name for representations; i.e. they are compatible with the mapping ρ that assigns to a G-set its associated permutation representation. ˆ is its com19.7. Almost ﬁnite G-sets. Given a group G its proﬁnite completion G pletion for the topology of normal subgroups of ﬁnite index. Or, equivalently it is the projective limit ˆ = lim G/N, G ←−

N runs over the normal subgroups of ﬁnite index

(19.8)

A G-set X for a group G is almost ﬁnite if for each x ∈ X the stabilizer subgroup Gx is a subgroup of ﬁnite index (so that all orbits are ﬁnite) and such that the subsets of invariants XU = {x ∈ X : Gx ⊆ U}

(19.9)

are ﬁnite for every subgroup of ﬁnite index. (So that there are only ﬁnitely many orbits of isomorphism type G/U for each such U.) It does not matter whether one works in this with the group G or its proﬁnite completion. In the case of a proﬁnite group one works with G-spaces. These are Hausdorff spaces with a continuous action97 of G. See [115], p. 5. Most papers that deal with the present subject (Burnside rings and Witt-Burnside rings) take the proﬁnite point of view. For a subgroup of ﬁnite index H ⊂ G restriction takes almost ﬁnite G-sets into almost ﬁnite H-sets and induction takes almost ﬁnite H-sets into almost ﬁnite G-sets. Finite disjoint unions and ﬁnite Cartesian products of almost ﬁnite G-sets are almost ﬁnite. 19.10. Burnside theorem. [78] Chapter XII theorem 1. Let G be a ﬁnite group and X and Y ﬁnite G-sets. Then the following are equivalent (i) The G-sets X and Y are isomorphic (ii) For any subgroup H of G the sets of invariants XH and Y H have the same cardinality. This still holds for almost ﬁnite G-sets where one only need consider subgroups of ﬁnite index. ˆ 19.11. Burnside ring. The (completed) Burnside ring B(G) of a group is now deﬁned as the Grothendieck ring of the category of almost ﬁnite G-sets. It can also be deﬁned as the projective limit of the usual Burnside rings (of ﬁnite groups) B(G/N) where N ranges over all normal subgroups (resp. closed normal subgroups) of ﬁnite index.98 97 Here G is given the topology deﬁned by the normal subgroups of ﬁnite index (which deﬁne the proﬁnite structure) and the set X is given the discrete topology. 98 This explains the notation B ˆ and the terminology ‘completed.’

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19.12. Almost ﬁnite cyclic sets. In the case of the group of integers all this specializes as follows. A cyclic set is simply a set with a left action of the group of integers on it. That is it is a set with a speciﬁed bijection. A cyclic set is almost ﬁnite if every orbit is of ﬁnite length and if for every n ∈ N there are only ﬁnitely many orbits of length n. Obviously (ﬁnite) disjoint unions and (ﬁnite) Cartesian products of almost ˆ ﬁnite cyclic sets are again almost ﬁnite and so there is the Grothendieck ring B(Z) of almost ﬁnite cyclic sets. ˆ If B(Z) denotes the Grothendieck ring of ﬁnite cyclic sets, B(Z) is the completion of B(Z) under a suitable natural topology on B(Z), see [120], p. 3 and below. ˆ 19.13. A remarkable commutative diagram. The W(Z), B(Z), (Z), Nr(Z) isomorˆ phisms. The ring B(Z) ﬁts in the following commutative diagram.

⌳( B(Z))

Nr(Z) itp

W(Z)

T

B(Z) ␸

w

ZN = Gh(Z)

id

ZN = Gh(Z)

SyP

syP

s

L(␸z)

⌳(Z)

(19.14)

t d log dt

coeff

tZ[[t]]

All the horizontal arrows in this diagram are isomorphisms (of rings with operators) and so is the morphism denoted ‘itp’ (which stands for ‘interpretation’). The morphisms w, ϕ, ˆ s, t dtd log are ghost component morphisms and injective. Finally, ϕZ and (ϕZ ) are surjections and SyP is an injection. Those morphisms which have not already occurred in earlier sections, such as w, s and the logarithmic derivative, will be elucidated below of course. One of the more remarkable aspects of this diagram is the fact that the composite syP ◦ T precisely embodies the very nasty coordinate change from Witt vector coordinates to power series coordinates encoded by the power series identity

1 = 1 + a1 t + a 2 t 2 + · · · (19.15) n) (1 − x t n n I consider this a main contribution from [120]. (The necklace coordinates also ﬁt in as suggested by the diagram.)

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ˆ 19.16. The ghost component morphism ϕˆ : B(Z) −→ Gh(Z) = ZN . Given an almost ﬁnite cyclic set X and an element n ∈ N, consider the subgroup of ﬁnite index nZ ⊆ Z and deﬁne ϕnZ (X) = #{x ∈ X : x is invariant under n Z}

(19.17)

ˆ This induces ring morphisms B(Z) −→ Z and these combine to deﬁne the ghost component ring morphism ϕˆ : Bˆ −→ Gh(Z). The morphism ϕˆ is injective because of the (extension to almost ﬁnite sets of the) Burnside theorem 19.10. Take the restrictions of the ϕnZ to B(Z) and give B(Z) the coarsest topology for ˆ which all these restrictions are continuous (with Z discrete). Then B(Z) is the completion of B(Z) for this topology. The morphism ϕˆ is injective; it is not surjective; an element y ∈ Gh(Z) is in its image if and only if μ(d −1 n)y(d) ≡ 0 mod n (19.18) d|n

for all n ∈ N. The same condition turned up in connection with the necklace ring in section 17 above ([113, 118, 126], [391] pp. 11–12). ˆ 19.19. The isomorphism itp: Nr(Z) −→ B(Z). The only transitive almost ﬁnite ˆ cyclic sets are the coset spaces Cn = Z/nZ. So every element of B(Z) is a (possibly inﬁnite) sum bn Cn , bn ∈ Z (19.20) n

As a group Nr(Z) = ZN with coordinate wise addition. So assigning to an element β = (b1 , b2 , b3 , . . .) ∈ ZN the formal difference of almost ﬁnite cyclic sets bn Cn − (−bn )Cn bn >0

bn <0

ˆ deﬁnes an isomorphism ‘itp’ of Abelian groups Nr(Z) −→ B(Z). It remains to see how itp behaves with respect to multiplication. Now observe99 that the product of two transitive cyclic sets Cr and Cs decomposes as the sum of (r, s) copies of C[r,s] . Here (r, s) is the greatest common divisor of r and s and [r, s] is their least common multiple. Given the deﬁnition of the multiplication on the necklace ring, see 17.15, it follows that itp is an isomorphism of rings. ˆ 19.21. The isomorphism T : W(Z) −→ B(Z). This isomorphism of rings is denoted τ in [120] and called “Teichmüller” there. This may be (indeed, is) appropriate in some sense but is also potentially confusing in view of the Teichmüller representatives mapping τ : A −→ W(A) in Witt vector theory. 99 It appears, see [120] p. 5, that it was this observation that lead to the investigations of Andreas Dress and Christian Siebeneicher that culminated in [115, 120, 126].

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The deﬁnition of the isomorphism T involves induction. Let X be a Z-set. Via the isomorphism Z −→ nZ ⊂ Z, r → nr it can be seen as an nZ-set. Now induce100 this one up to Z. The result is denoted indn (X). In concreto this works out as follows. Consider the product Z × X and the equivalence relation on it deﬁned by (z, x) ∼ (z − nu, ux), u ∈ Z. Then indn (X) is the set of equivalence classes of this equivalence relation with the action induced by the left action of Z on itself, (z , (z, x)) → (z + z, x). It readily follows that indn (Cr ) = Crn and hence that ind n

br Cr

=

r

where

br

=

br/n 0

r

br Cr

(19.22)

if n divides r otherwise

and hence that ϕrZ (ind n (X)) =

nϕr/n (X) 0

(19.23)

if n divides r otherwise

(19.24)

So, at the ghost component level indn behaves just like Verschiebung in the case of the big Witt vectors. Now let q ∈ N∪{0}. With q(z) denote the set of maps Z −→ {1, 2, · · · , q} that factor through some set of cosets Z / nZ. That is the continous maps Z −→ {1, 2, · · · , q} where {1, 2, · · · , q} has the discrete topology and Z the topology of subgroups of ﬁnite index. The action of Z is given by (zf )(z ) = f(z − z)

(19.25)

Note that 0(Z) = Ø. It is fairly immediate that (qq )(Z) = q(Z) × q (Z) , ϕnZ (q(Z) ) = qn

(19.26)

ϕ(q ˆ (Z) ) = (1, q, q2 , q3 , . . .) = coeff (1 − q)−1

(19.27)

Finally deﬁne ˆ T : W(Z) −→ B(Z), (x1 , x2 , x3 , · · · ) →

∞ n=1

100 See 19.4 above.

ind n xn(Z)

(19.28)

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Then it follows from what has been noted just above that n/d ϕnZ (x1 , x2 , x3 , · · · ) = dxd

(19.29)

d|n

and that ∞

(1 − xn t n )−1 =

n=1

∞

(1 − t n )−bn ⇔ T(x1 , x2 , x3 , · · · ) =

n=1

∞

bi Ci

(19.30)

i=1

Formula (19.29) shows that the lower left square in the diagram (19.14) is commutative. Using the characterizations of the image of the ghost components morphisms w and ϕ, ˆ see (9.95) and (19.20) above, and their injectivity, it follows that T is an isomorphism. Further (19.30) says that the composite morphism (itp)−1 ◦ T exactly embodies the coordinate transformation between Witt vector coordinates and necklace coordinates encoded by the power series identity ∞

∞

(1 − xn t n )−1 =

n=1

(1 − t n )−bn

n=1

19.31. Symmetric powers of G-sets. Given a G-set X its n-th symmetric power is obtained by ﬁrst taking the n-fold Cartesian product Xn with diagonal action g(x1 , x2 , · · · , xn ) = (gx1 , gx2 , · · · , gxn )

(19.32)

The symmetric group S n acts on this by permuting coordinates σ(x1 , x2 , · · · , xn ) = (xσ −1 (1) , xσ −1 (2) , · · · , xσ −1 (n) ) and this action commutes with the diagonal action of G. So there is an induced action of G on the set of orbits S n X. This is the n-th symmetric power of the G-set X. Alternatively, and better, an element from S n X canbe described as a multiset of size n, i.e. a function f : X −→ N ∪ {0} such that x∈X f(x) = n. In this picture the action is (gf )(x) = f(g−1 x). The symmetric powers of almost ﬁnite G-sets are almost ﬁnite. It follows rather immediately from the construction that S n (X Y ) = S i X × S j Y i+j=n

(19.33)

and it follows that the map X →

∞

(S n X)t n

(19.34)

n=1

induces a morphism of Abelian groups ˆ ˆ SyP : B(Z) −→ (B(Z))

(19.35)

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449

ˆ Now compose this with (ϕz ) : (B(Z)) −→ (Z) to obtain a morphism of Abelian groups ˆ syP : B(Z) −→ (Z)

(19.36)

(and the commutative upper triangle of diagram (19.14)). Thus the coefﬁcient of t n in syP(X) is the number of invariant elements in the n-th symmetric power S n X. It now turns out that in fact syP is an isomorphism of rings and even an isomorphism of λ−rings. The key to that is the easy observation that & 1 if r divides n n ϕz (S Cr ) = (19.37) 0 otherwise (Indeed, a function f : {0, 1, · · · , r − 1} −→ N ∪ {0} is invariant under ‘left shift modulo r of its argument’ if and only if all its values are equal.) Formula (19.37) serves to prove that the lower right square of diagram (19.14) is commutative (as well as the two triangles there, that syP is an isomorphism of rings, and that the composed morphism syP ◦ itp embodies the coordinate change encoded by the power series identity ∞

(1 − t n )−bn = 1 + a1 t + a2 t 2 + a3 t 3 +

(19.38)

n=1

When combined with (19.30) this shows that the composite morphism syP ◦ T : W(Z) −→ (Z) precisely gives the nasty coordinate change formulas between Witt vector coordinates and power series coordinates encoded by the power series identity ∞

(1 − xn t n )−1 = 1 + a1 t + a2 t 2 + a3 t 3 + · · ·

n=1

which I feel is a major insight from [120]. ˆ 19.39. Frobenius and Verschiebung on W(Z), B(Z), (Z). On all the rings in diagram (19.14) there are Frobenius and Verschiebung operators. All have been ˆ deﬁned before except the ones on B(Z). On this Burnside ring ‘Verschiebung’ is ‘ind’ and Frobenius is restriction (followed by the identiﬁcation coming from the obvious (canonical) isomorphism of Abelian groups Z −→ nZ. It now turns out (and is veriﬁed with no great difﬁculty) that all morphism indicated in the diagram are compatible with the Frobenius operators and all except SyP are compatible with Verschiebung. (SyP is most deﬁnitely not compatible with Verschiebung.) ˆ As a matter of fact, identifying B(Z) with (Z) via syP turns SyP into the morphism of λ-rings σt : (Z) −→ ((Z)) of λ-ring theory,101 see section 16 above. 101 This is one more indication that it is better to work with symmetric powers than with exterior powers.

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19.40. This concludes the treatment here of the Burnside ring of the integers (or ˆ I have pretty much followed [120] apart from some its proﬁnite completion Z). interspersed remarks. I hope and believe that the outline above is sufﬁcient that the reader can ﬁll in all details. But if needed they can be found in loc. cit. However, there is more to the Burnside ring story in connection with Witt vectors. For every proﬁnite group there is a Witt vector like functor CRing −→ CRing called the Witt Burnside functor. Below there is the main theorem from [126] about them. First some deﬁnitions and notation. 19.41. Let G be a proﬁnite group. That is a projective limit G = lim G/N of ﬁnite ←−

quotient group. Give G the topology deﬁned by the collection of normal subgroups N of ﬁnite index. A G-space is a Hausdorff spcae on which G acts continuously from the left. Such a G-space is almost ﬁnite if it is discrete and if for any open subgroup U ⊂ G the number of invariants under U is ﬁnite. Set ϕU (X) = #{x ∈ X: ux = x for all u ∈ U}

(19.42)

For any open subgroup U there is the transitive almost ﬁnite (in fact ﬁnite) G-space of left cosets G/U. Denote with osg(G) the set of all open subgroups of G and with cosg(G) the quotient of conjugacy classes of open subgroups. ˆ Finally let B(G) be the “completed Burnside ring” of G, that is the Grothendieck ring of (isomorphism classes) of almost ﬁnite G-spaces with addition induced by disjoint union and multiplication induced by the Cartesian product of G-spaces. 19.43. Existence theorem of the Witt-Burnside functors [126]. Let G be a proﬁnite group. There exists a unique covariant functor WG from the category of unital commutative rings to itself such that as a set WG (A) is the set Acosg(G) of all functions from cosg(G) to A. That is all functions from osg(G) to A that are constant on conjugacy classes of subgroups, and with WG (h): WG (A) −→ WG (B) given by composition α → α ◦ h, α ∈ WG (A), such that for all open subgroups the map A ψU : WG (A) −→ A

(19.44)

deﬁned by

ϕU (G/V )α(V )(V :U )

(19.45)

U⊂scjg V ⊂G

is a natural transformation of functors from WG to the identity. Here the sum (9.45) is over all V such that U is subconjugate to U (denoted U ⊂scjg V ), which means that there is a g ∈ G such that U is a subgroup of gVg−1 , the ‘index’ (V :U ) = (G:U )/(G:V ), and in the sum (9.45) exactly one summand is taken for each conjugacy class of subgroups V with U ⊂scjg V . ˆ Moreover, WG (Z) = B(G) and Wz = W the functor of the big Witt vectors. There are also Frobenius and Verschiebung like functorial endomorphisms coming respectively from restriction and induction. The Frobenius morphism are ring

Witt vectors. Part 1

451

morphisms, the Verschiebung morphism are morphisms of Abelian groups. These morphisms have the usual kinds of properties. Since the appearance of the foundational papers [126, 120] a number of other papers have appeared on the topic of Witt-Burnside functors, giving reﬁnements, further developments, applications and interrelations, simpliﬁcations and complications; see [73, 116, 122, 135, 156, 175, 318, 319, 315]. 19.46. β-rings. It seems clear from [370] that there is no good way to deﬁne a λ-ring structure on Burnside rings, see also [158]. There are (at least) two different choices giving pre-λ-rings but neither is guaranteed to yield a λ-ring. Of the two the symmetric power construction seems to work best. Instead one needs what are called β-operations102 (power operations), ﬁrst introduced in [56], and β-rings. There is a β-operation for each conjugacy class of subgroups of the symmetric groups Sn and they are constructed by means of symmetric powers of G-sets. Every λ-ring is a β-ring (but not vice versa). There is some vagueness about what is precisely the right deﬁnition of a β-ring, see, e.g. the second paragraph on page 2 of [183] and the second comment at the end of §3 of that preprint. However, it seems clear that the free β-ring on one generator must be the ring ∞

B(S ) = ⊕ B(Sn ) n=0

(19.47)

This is the direct sum of the Burnside rings of the symmetric groups (with B(S0 ) = Z by decree) equipped with an outer product deﬁned completely analogously as in the case of R(S ), see section 18 above. I.e. S

(X × Y ) XY = IndSi+j i ×Sj

(19.48)

There is also a coproduct making it a Hopf algebra and the underlying ring is again a ring of polynomials in countably many indeterminates over the integers. The natural morphism ρ : B(S ) −→ R(S)

(19.49)

that assigns to an Sn -set the corresponding permutation representation is a surjective (but not injective) morphism of Hopf algebras and also a morphism of λ-rings, see [183] p. 11. A selection of papers on β-rings is [56, 158, 183, 300, 301, 344, 397, 396, 433]. Coda. Some people write about the Witt polynomials as mysterious polynomials that come out of nowhere.103 To me they are so elegant and natural that they simply cry out for 102 I have tried to track down where the appellation ‘beta’ comes from unsuccessfully. But it seems likely that it is some sort of philosophical mix between the ‘B’ from Burnside and the ‘λ’ from λ-ring. 103 Also, see section 3 above, they deﬁnitely do not come out nowhere.

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deep study. This is my main reason for presenting things here as I have done above, even though, as has been shown, they are not really needed. Should a reader inadvertently get really interested in the Witt vector ring functors he/she is recommended to work through the 57 exercises on the subject in [63], pp. 26 worth, just for the formulations only, mostly contributed, I have been told, by Pierre Cartier. He/she can then continue with the exercises on Cohen rings in same volume which also involve a fair amount of Witt vector stuff. Appendix. The algebra of symmetric functions in inﬁnitely many indeterminates In several places in the sections above there occur expressions like ∞

i=1

1 + ξi t + ξi2 t 2 + ξi3 t 3 + · · ·

(A.1)

and statements that the coefﬁcients of each power in t in (A.1) are symmetric functions in the inﬁnity of commuting variables ξ1 , ξ2 , ξ3 , · · · and that they are, hence, polynomials in the elementary (or complete) symmetric functions in the inﬁnity of commuting indeterminates ξ1 , ξ2 , ξ3 , . . .. The few pages in this appendix are meant for those who feel (rightly) a bit nervous about statements like this even though the meaning seems intuitively clear.104 A.2. Power series in inﬁnitely many variables. Let I be any set and let {ξi : i ∈ I} be a corresponding set of commuting indeterminates. An exponent sequence for I is a map e : I −→ N ∪ {0} of ﬁnite support. I.e. there are only ﬁnitely may i ∈ I for which e(i) = 0. Let E(I ) be the set of all exponent sequences. For each e ∈ E(I ) introduce a symbol ξ e . Giving I a total order, one can think of ξ e as

e(i) ξe = ξi (A.3) e(i)=0

where the product (monomial) on the right hand side is written down in the order speciﬁed by an ordering of I.105 Two monomials, i.e. symbols, ξ e are multiplied by

the rule ξ e ξ e = ξ e+e where e + e is the point-wise sum (e + e )(i) = e(i) + e (i). The ring of formal power series over a base ring k is now deﬁned as ⎧ ⎫ ⎨ ⎬ ae ξ e: ae ∈ k (A.4) kξi : i ∈ I = ⎩ ⎭ e∈E(I )

104 To my mind the matters touched upon here illustrate well issues in foundational mathematics (intuitionism) having to do with realized inﬁnities vs potential inﬁnites. 105 But this is not needed.

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Two such expressions as on the right of (A.4) are multiplied by the rule ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎝ ae ξ e ⎠ ⎝ be ξ e ⎠ = ⎝ ce ξ e ⎠, ce = ae be

(A.5)106 e∈E(I )

e∈E(I )

e +e

= e

e∈E(I )

This product rule is well deﬁned because of the ﬁnite support condition on exponent sequences. A product like (A.1) is now by deﬁnition interpreted as the formal power series one obtains by multiplying any ﬁnite number of factors from it. That is ⎛ ⎞ ∞

⎝1 + ai,ei ξ ei t wt(ei ) ⎠ = 1 + ce ξ e t wt(e), i=1

wt(e) =

wt(ei )≥1

i

e(i),

wt(e)≥1

ce =

(A.6)

ai1 ,e1 · · · air ,er

e1 +···+er =e wt(ei )≥1

and similar products with the ‘counting variable’ t left out. A.7. Symmetric and quasisymmetric power series. Probably every one (who is likely to get his ﬁngers on this chapter) knows, or can easily guess at, what is a symmetric polynomial (over the integers, or any other base rink k). Say, the polynomial is in n (commuting) variables ξ1 , ξ2 , · · · , ξn . Then the polynomial f = f(ξ1 , ξ2 , · · · , ξn ) is symmetric if for every permutation σ of {1, 2, . . . , n} f(ξσ(1) , ξσ(2) , · · · , ξσ(n) ) = f(ξ1 , ξ2 , · · · , ξn )

(A.8)

And, in that case, the main theorem of symmetric functions says that: the polynomial f is a polynomial pf in the complete symmetric functions h1 , h2 , · · ·, hn or, equivalently, in the elementary symmetric functions, and “the polynomial pf is independent of the number of variables involved provided there are enough of them” (meaning more than or equal to the degree of f ). The last phrase needs explaining. Also this strongly suggests that the best way to work with symmetric polynomials is to take an inﬁnity of variables. A little bit more notation is useful. Let SI be the group (under composition) of all bijections σ: I −→ I such that there are only ﬁnitely many i ∈ I with σ(i) = i. For SN the notation S∞ is usually used. For an exponent sequence e let eσ be the exponent sequence eσ (i) = e(σ −1 (i)).107 106 All this just formalizes what everyone knows intuitively. One can also do these things for noncommuting indeterminates, and things are (curiously enough?) actually easier in that case. For totally ordered index sets one can go much further and make sense of inﬁnite ordered products and sums by using injective limit ideas to make sense of inﬁnite ordered sums and in ﬁnite products of elements of the base ﬁeld and the integers by treating them as sequences with two sequences equal if they eventually agree. 107 The exponent ‘−1’ is there to ensure that eστ = (eσ )τ ; not that that is important in the present context.

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A power series

e∈E(I )

ae ξ e is now symmetric if and only if ae = aeσ for all σ ∈ SI .

A polynomial in an inﬁnite set of commutating indeterminates is a ﬁnite sum ae ξ e i.e. a power series with all but ﬁnitely many of the coefﬁcients ae unequal e∈E(I ) ae ξ e is of bounded degree if and only if there is a natural to zero. A power series e∈E(I )

number n such that ae = 0 for all e with wt(e) > n. The complete symmetric functions in the ξ are by deﬁnition hn = ξe (A.9) wt(e)=n

The main theorem for symmetric power series in an inﬁnity of indeterminates now takes the following form. A.10. Theorem. Every symmetric power series can be uniquely written as a power series in the complete symmetric functions. A bounded degree symmetric power series is a polynomial in the complete symmetric functions. There is of course a similar theorem in terms of the elementary symmetric functions. The algebra Symm is the algebra of polynomials in the countable set of complete symmetric functions hn for the case of the index set I = N. A.11. Projective limit description. For the ﬁnicky (or pernickety) the following projective limit construction is perhaps more congenial. For each n consider the algebra morphism πn+1,n: k [ξ1 , · · ·, ξn , ξn+1 ] −→ k [ξ1 , · · ·, ξn ], ξi → ξi for i ∈ {1, · · ·, n}, ξn+1 → 0 These are graded algebra morphisms (n)

Let Symmk be the subalgebra of symmetric polynomials k [x1 , · · · , xn ]. This gives a graded projective system of graded algebras (n+1)

πn+1,n: Symmk

(n)

−→ Symmk

and Symm is the projective limit of this system. So, for instance, the symmetric graded ξ e is not in Symm. In this picture theorem A.10 is obtained by the power series e∈E(N)

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Crystal Graphs and the Combinatorics of Young Tableaux Jae-Hoon Kwon Department of Mathematics, University of Seoul, Seoul 130-743, Korea E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Quantum group and crystal base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Quantum group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Crystal base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Crystals and Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Crystal of words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Crystal of a Young tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Crystal of a rational Young tableau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Crystal equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Knuth equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Robinson–Schensted correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Littlewood–Richardson rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Bicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Robinson–Schensted–Knuth correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Dual Robinson–Schensted–Knuth correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Decomposition of Fock space representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00208-8 473

474 474 474 476 480 480 481 482 484 486 486 488 489 491 491 495 497 503

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1. Introduction The quantum group Uq (g) of a symmetrizable Kac–Moody algebra g [17] is a q-analogue of the universal enveloping algebra U(g), which was introduced independently by Drinfel’d [7] and Jimbo [15], and it has been one of the most important objects in representation theory in connection with many areas of mathematics and mathematical physics. In [21, 22], Kashiwara introduced a crystal base of an integrable representation of Uq (g). A crystal base is usually understood as a basis at q = 0, or a crystallized basis because the parameter q in the quantum group Uq (g) corresponds to the absolute temperature in certain exactly solvable lattice models in statistical mechanics. Remarkably, to a crystal base, we can associate a unique colored oriented graph called a crystal graph, which reﬂects the combinatorial structure of a given integrable representation. Therefore, with crystal base theory, many problems in representation theory are reduced to problems in combinatorics, and also certain combinatorial results naturally obtain representation theoretical meanings (interpretations). On the other hand, the notion of Young tableaux together with its combinatorics plays an essential role in the representation theory of the general linear Lie algebra gln or the symmetric group Sk . There are many beautiful algorithms by which we understand the combinatorics of Young tableaux, based on the Schensted bumping rule, the Schützenberger sliding rule, and the Knuth equivalence. Among them, the Robinson– Schensted–Knuth correspondence and the Littlewood–Richardson rule are the most important ones closely related to representation theory (see, for example, [10,41,43]). The purpose of this survey article is to give an exposition on the relations between the classical combinatorics of Young tableaux and the crystal graphs of integrable Uq (gln )-modules, as an introduction to crystal base theory for those who are not familiar with this area (see also [5] that deals with similar topics, but is written from a different point of view). This article is organized as follows. In Section 2, we recall the deﬁnition of the quantum group Uq (gln ), its integrable representations, and their crystal bases. In Section 3, we review the Kashiwara and Nakashima description of the crystal graphs of integrable highest weight Uq (gln )-modules [26]. In Section 4, we explain the Robinson–Schensted correspondence and the Littlewood–Richardson rule in the context of crystal graphs. Our explanation is based on the crucial fact that the Knuth equivalence on words is a special case of the crystal equivalence. In Section 5, we discuss the Robinson–Schensted–Knuth correspondence in a similar vein, and give another proof of it using the notion of crystals. We also give some applications to combinatorics and representation theory. 2. Quantum group and crystal base 2.1. Quantum group For n ≥ 2, let Pn = 1≤i≤n Zi be the free abelian group with a basis {1 , . . . , n }1 , called the weight lattice. There is a natural symmetric bilinear form ( , ) on Pn given 1 , . . . , are formal symbols. n 1

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by (i , j ) = δij for 1 ≤ i, j ≤ n. Let , denote the canonical pairing on Pn∨ × Pn , where Pn∨ = Hom(Pn , Z) is the dual weight lattice. Set In = {1, . . . , n − 1}. Then, for i ∈ In the simple root αi is given by αi = i − i+1 , and the simple coroot hi is ∨ deﬁned to be the unique element in Pn such that hi , λ = (αi , λ) for all λ ∈ Pn . Note that hj , αi i,j∈I = (αi , αj ) i,j∈I is the Cartan matrix of An−1 type (see [17]). n n Let gln be the general linear Lie algebra over a ﬁeld K of characteristic 0 and let q be an indeterminate. We deﬁne Uq (gln ) to be the associative algebra over K(q) with 1 generated by qh , ei , and fi (h ∈ Pn∨ and i ∈ In ) subject to the following relations; q0 = 1,

qh+h = qh qh ,

qh ei q−h = qh,αi ei , ei fj − fj ei = δij

qh fi q−h = q−h,αi fi ,

qhi − q−hi , q − q−1

ei ej = ej ei , fi fj = fj fi ,

if |i − j| > 1,

ej e2i − (q + q−1 )ei ej ei + e2i ej = 0, fj fi2 − (q + q−1 )fi fj fi + fi2 fj = 0,

if |i − j| = 1,

for h, h ∈ Pn∨ and i, j ∈ In . We call Uq (gln ) the quantized universal enveloping algebra of gln (or simply, the quantum group of gln ). Also, we deﬁne Uq (sln ) to be the subalgebra of Uq (gln ) generated by ei , fi and q±hi for i ∈ In . Note that for each i ∈ In , Uq (gln )i , the subalgebra of Uq (gln ) generated by ei , fi , q±hi is isomorphic to Uq (sl2 ). A left Uq (gln )-module M is called integrable if ei and fi act locally nilpotently on M. Let Oint be the category of integrable Uq (gln )-modules M satisfying the following conditions; (1) M has a weight space decomposition, that is, M = λ∈Pn Mλ , where Mλ = { u ∈ M | qh u = qh,λ u for all h ∈ Pn∨ }, and dimK(q) Mλ < ∞ for λ ∈ Pn (λ is called a weight of M if Mλ = {0}). (2) The weights of M are dominated by a ﬁnite number of elements in Pn , that is, there exist λ1 , . . . , λr ∈ Pn such that every weight of M is contained in λk − i∈In Z≥0 αi for some 1 ≤ k ≤ r. Note that any M in Oint is a direct sum of ﬁnite dimensional representations of Uq (gln )i Uq (sl2 ) for each i ∈ In . The representation theory of Uq (gln ) is almost parallel to that of its Lie algebra gln . Set Pn+ = { λ ∈ Pn | hi , λ ≥ 0 for i ∈ In }. An element in Pn+ is called a dominant integral weight. We may identify Pn with Zn , and hence Pn+ with Zn+ = { λ = (λ1 , . . . , λn ) ∈ Zn | λi ≥ λi+1 for 1 ≤ i ≤ n − 1 }. There is a partial ordering ≤ on Pn such that μ ≤ λ if and only if λ−μ ∈ i∈In Z≥0 αi for λ, μ ∈ Pn .

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For λ ∈ Pn+ , let V(λ) be the Uq (gln )-module generated by vλ 2 subject to the following relations; qh vλ = qh,λ vλ ,

ei vλ = 0,

1+hi ,λ

fi

vλ = 0,

for h ∈ Pn∨ and i ∈ In . Then V(λ) is an irreducible Uq (gln )-module, and any Uq (gln )that are isomodule M in Oint is completely reducible with irreducible summands morphic to V(λ) for some λ ∈ Pn+ , that is, M is isomorphic to λ∈Pn+ V(λ)⊕mλ for some mλ ∈ Z≥0 (see [36]). We call V(λ) the irreducible highest weight module with highest weight λ, and vλ the highest weight vector, since for each weight μ of V(λ), we have μ ≤ λ. The quantum group Uq (gln ) has a Hopf algebra structure, where the comultiplication : Uq (gln ) → Uq (gln ) ⊗ Uq (gln ) is given by qh = qh ⊗ qh , ei = ei ⊗ q−hi + 1 ⊗ ei , fi = fi ⊗ 1 + qhi ⊗ fi , for h ∈ Pn∨ and i ∈ In . So, we can deﬁne a Uq (gln )-module structure on the tensor product of two Uq (gln )-modules. In particular, the category Oint is closed under tensor product. Remark 2.1. For more detailed expositions on quantum groups of symmetrizable Kac–Moody algebras and their representations, see the books [3, 12, 14, 36]. 2.2. Crystal base Let us brieﬂy review the crystal base theory developed by Kashiwara (see also [24] for a general review on crystal bases). For M ∈ Oint and i ∈ In , M is a direct sum of ﬁnite dimensional irreducible representations of Uq (gln )i . Then each u ∈ Mλ (λ ∈ Pn ) can be written uniquely as (k) u= fi uk , k≥0,−hi ,λ

with ei uk = 0 for all k ≥ 0, where [a] = Then we deﬁne e˜ i u =

qa − q−a (a ≥ 1), q − q−1 k≥1

(k−1)

fi

uk ,

[k]! =

f˜i u =

k

[a],

a=1

k≥0

(k+1)

fi

(k)

fi

=

1 k f . [k]! i

uk .

2 More precisely, V(λ) = U (gl )/ U (gl )e + U (gl )(qh − qh,λ ) + U (gl )f 1+hi ,λ , and v q λ n n i n n i i q h q i q

is the image of 1 ∈ Uq (gln ) in V(λ).

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The operators e˜ i , f˜i : M → M are called Kashiwara operators. Let A be the subring of K(q) that consists of rational functions regular at q = 0. A free A-module L is called a crystal lattice of M if (1) L is an A-lattice of M, that is, M = K(q) ⊗A L, (2) L = λ∈Pn Lλ , where Lλ = Mλ ∩ L, (3) e˜ i L ⊂ L and f˜i L ⊂ L for i ∈ In . Note that the Kashiwara operators also induce K-linear endomorphisms on L/qL. Now, a crystal base of M is a pair (L, B) satisfying the following conditions; (1) (2) (3) (4) (5)

L is a crystal lattice of M, B is a K-basis of L/qL,

B = λ∈Pn Bλ , where Bλ = (Lλ /qLλ ) ∩ B, e˜ i B ⊂ B ∪ {0} and f˜i B ⊂ B ∪ {0} for i ∈ In . for b, b ∈ B and i ∈ In , f˜i b = b if and only if e˜ i b = b.

It is not difﬁcult to see that dimK(q) Mλ = |Bλ | for λ ∈ Pn . Moreover, to a crystal base (L, B) of M, we can associate an In -colored oriented graph, where the set of vertices is given by B and the colored arrows are determined by i

b → b

if and only if b = f˜i b

(i ∈ In ).

We call B together with the In -colored oriented graph structure the crystal graph of M. Theorem 2.2 (Existence of a crystal base [21, 22]). For λ ∈ Pn+ , let Af˜i . . . f˜i vλ , L(λ) = 1

r≥0, i1 ,...,ir ∈In

B(λ) = { f˜i1 . . . f˜ir vλ

r

(mod qL(λ)) | r ≥ 0, i1 , . . . ir ∈ In }\{0}.

Then (L(λ), B(λ)) is a crystal base of V(λ). It follows directly from the theorem aforementioned that any M ∈ Oint has a crystal base. That is, if M is isomorphic to λ∈Pn+ V(λ)⊕mλ for some mλ ∈ Z≥0 ,

then a crystal base of M can be given by ( λ∈Pn+ L(λ)⊕mλ , λ∈Pn+ B(λ)⊕mλ ). Theorem 2.3 (Uniqueness of a crystal base [21, 22]). For M ∈ Oint , let (L, B) be a crystal base of M. Then there exists an isomorphism of Uq (gln )-modules V(λ)⊕mλ ψ : M −→ λ∈Pn+ + for some mλ ∈ Z≥0⊕m(λ ∈ Pn ) such that ψ restricts to an A-linear isomorphism from λ L to λ∈Pn+ L(λ) , and the induced K-linear map ψ gives a bijection from B to

⊕mλ . + B(λ) λ∈Pn

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Note that the crystal graph B(λ) is connected, and any element in B(λ) is generated by the highest weight element vλ = vλ + qL(λ) ∈ L(λ)/qL(λ), so that for any b ∈ B(λ), we have b = f˜i1 . . . f˜ir vλ for some i1 , . . . , ir ∈ In . Also, the highest weight element in B(λ) can be characterized by the fact that e˜ i vλ = 0 for all i ∈ In . By the uniqueness theorem, there exists a unique In -colored oriented graph B associated with each M ∈ Oint , where each connected component in B is isomorphic to B(λ) for some λ ∈ Pn+ as an In -colored oriented graph. Example 2.4. Let us consider crystal bases of integrable Uq (sl2 )-modules. Suppose that Uq (sl2 ) is generated by e, f , and q±h . Each dominant integral weight is determined by its value on h, which is a nonnegative integer. For m ≥ 0, let V(m) be the irreducible highest weight module with highest weight m, and vm the highest weight vector. Then dimK(q) V(m) = m + 1 with a basis { vm , fvm , . . . , f m vm }. If we put Af (k) vm , B(m) = { f (k) vm | 0 ≤ k ≤ m }, L(m) = 0≤k≤m

then (L(m), B(m)) is a crystal base of V(m). The associated crystal graph of V(m) is given by vm −→ fvm −→ · · · −→ f (m) vm . Note that the weight of f (k) vm is m − 2k for 0 ≤ k ≤ m. For M ∈ Oint , let (L, B) be a crystal base of M. For each i ∈ In , (L, B) is also a crystal base of M as a representation of Uq (gln )i Uq (sl2 ). Hence, the connected component of each b in B, as the crystal graph of a Uq (sl2 )-module, is equal to the crystal graph of V(m) for some m ≥ 0, called the i-string of b. Set εi (b) = max{ k ∈ Z≥0 : e˜ ki b = 0 }, k ϕi (b) = max{ k ∈ Z≥0 : f˜i b = 0 },

which denote the number of i-arrows coming into b and going out of b in the i-string of b, respectively3 . Then we have ϕi (b) − εi (b) = hi , wt(b), where wt(b) denotes the weight of b. One of the most important features of a crystal base is that it has a nice behavior under tensor product, which can be stated as follows; Theorem 2.5 (Tensor product of crystal bases [21, 22]). For Mi ∈ Oint , let (Li , Bi ) be the crystal base of Mi (i = 1, 2). Then (L1 ⊗ L2 , B1 ⊗ B2 ) is a crystal base of 3 Note that ε is a function on B, and is a basis element of P . n i i

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479

M1 ⊗ M2 . Moreover, for b1 ⊗ b2 ∈ B1 ⊗ B2 and i ∈ In , we have e˜ i b1 ⊗ b2 , if ϕi (b1 ) ≥ εi (b2 ), e˜ i (b1 ⊗ b2 ) = b1 ⊗ e˜ i b2 , if ϕi (b1 ) < εi (b2 ),

f˜ b1 ⊗ b2 , f˜i (b1 ⊗ b2 ) = i b1 ⊗ f˜i b2 ,

if ϕi (b1 ) > εi (b2 ), if ϕi (b1 ) ≤ εi (b2 ).

The formulas mentioned earlier for the actions of e˜ i and f˜i are often called the tensor product rule, and this rule plays a crucial role when we construct various crystal graphs (see the next section). The tensor product rule also implies that under the same hypothesis, εi (b1 ⊗ b2 ) = max(εi (b1 ), εi (b2 ) − hi , wt(b1 )), ϕi (b1 ⊗ b2 ) = max(ϕi (b1 ) + hi , wt(b2 ), ϕi (b2 )). Example 2.6. Let us illustrate the tensor product rule for the crystal bases of Uq (sl2 )-modules. We keep the notations of the previous example. Let B(3) = { bk = f (k) v3 | k = 0, 1, 2, 3 } and B(2) = { bk = f (k) v2 | k = 0, 1, 2 } be the crystal graphs of V(3) and V(2), respectively. Then the crystal graph B(3)⊗B(2) is given as follows; b0 ⊗ b0

−→ b1 ⊗ b0

−→ b2 ⊗ b0

b0 ⊗ b1

−→ b1 ⊗ b1

b0 ⊗ b2

−→ b1 ⊗ b2

−→ b2 ⊗ b1 ↓ b2 ⊗ b2

−→ b3 ⊗ b0 ↓ b3 ⊗ b1 ↓ b3 ⊗ b2

This implies that B(3) ⊗ B(2) = B(5) B(3) B(1), and hence V(3) ⊗ V(2) V(5) ⊕ V(3) ⊕ V(1). Remark 2.7. The results on crystal bases presented here were proved in case of the quantum group Uq (g) of a symmetrizable Kac–Moody algebra g by Kashiwara [22]. Also, it is an important problem to give an explicit realization of the crystal graph of a Uq (g)-module. There have been many papers on this problem: for classical Lie algebras, there are tableaux realizations by Kashiwara and Nakashima [26]. For afﬁne Lie algebras, there are path realizations using perfect crystals [20]. For a symmetrizable Kac–Moody algebra g, Littelmann realized the crystal graph of an irreducible integrable highest weight module in terms of certain piecewise linear paths in the weight lattice, often called the Littelmann path model [34]. Recently, Nakajima introduced the monomial realization, where the crystal graphs are given in terms of certain monomials in commuting variables [25, 39].

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3. Crystals and Young tableaux 3.1. Crystal Motivated by the colored oriented graphs induced from crystal graphs of integrable Uq (gln )-modules, we may deﬁne the notion of (abstract) crystals. Deﬁnition 3.1. ([23, 24]) Let Pn be the weight lattice, Pn∨ the dual weight lattice, { αi | i ∈ In } the set of simple roots, and { hi | i ∈ In } the set of simple coroots given in 2.1. A gln -crystal is a set B together with the maps wt : B → Pn , εi , ϕi : B → Z ∪ {−∞}, ei , fi : B → B ∪ {0}, for i ∈ In , satisfying the following conditions; (1) for i ∈ In and b ∈ B, we have ϕi (b) = hi , wt(b) + εi (b), (2) if ei b ∈ B for i ∈ In and b ∈ B, then εi (ei b) = εi (b) − 1,

ϕi (ei b) = ϕi (b) + 1,

wt(ei b) = wt(b) + αi ,

(3) if fi b ∈ B for i ∈ In and b ∈ B, then εi (fi b) = εi (b) + 1,

ϕi (fi b) = ϕi (b) − 1,

wt(fi b) = wt(b) − αi ,

(4) fi b = b if and only if b = ei b for all i ∈ In , b, b ∈ B, (5) If ϕi (b) = −∞, then ei b = fi b = 0, where 0 is a formal symbol and −∞ is the smallest element in Z ∪ {−∞} and is taken to satisfy −∞ + n = −∞ for all n ∈ Z. Remark 3.2. (1) A gln -crystal B is an In -colored oriented graph, where i

b → b

if and only if b = fi b

(i ∈ In ).

We also call ei and fi (i ∈ In ) Kashiwara operators. (2) For λ ∈ Pn+ , B(λ) is a gln -crystal with ei = ei and fi = fi for i ∈ In (see 2.2). In fact, the gln -crystals, which we deal with in this chapter, will always satisfy the following additional conditions; εi (b) = max{ k ∈ Z≥0 : eik b = 0 },

ϕi (b) = max{ k ∈ Z≥0 : fik b = 0 },

for b ∈ B and i ∈ In . Such crystals are called seminormal [24]. There are gln -crystals that are not isomorphic to a B(λ), but play important roles in the construction of various crystal graphs, for example the crystal graphs of the Verma modules and the negative part of a quantum group and so on (see [23]).

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Let B1 and B2 be gln -crystals. Let B1 ⊗ B2 be { b1 ⊗ b2 | bi ∈ Bi (i = 1, 2) } as a set. Deﬁne wt(b1 ⊗ b2 ) = wt(b1 ) + wt(b2 ), εi (b1 ⊗ b2 ) = max(εi (b1 ), εi (b2 ) − hi , wt(b1 )), ϕi (b1 ⊗ b2 ) = max(ϕi (b1 ) + hi , wt(b2 ), ϕi (b2 )), ei b1 ⊗ b2 , if ϕi (b1 ) ≥ εi (b2 ), ei (b1 ⊗ b2 ) = b1 ⊗ ei b2 , if ϕi (b1 ) < εi (b2 ), fi b1 ⊗ b2 , if ϕi (b1 ) > εi (b2 ), fi (b1 ⊗ b2 ) = b1 ⊗ fi b2 , if ϕi (b1 ) ≤ εi (b2 ), where we assume that 0 ⊗ b2 = b1 ⊗ 0 = 0. Then, it is straightforward to check that B1 ⊗ B2 is a gln -crystal with respect to the wt, εi , ϕi , ei , fi (i ∈ In ) on B1 ⊗ B2 , and it is called the tensor product of B1 and B2 . Finally, let us ﬁx some terminology. Let B be a gln -crystal. A subset B ⊂ B is called a subcrystal of B if B itself is a gln -crystal with respect to the wt, εi , ϕi , ei , fi (i ∈ In ) of B. Let B1 and B2 be two gln -crystals. The direct sum B1 ⊕B2 is the disjoint union of B1 and B2 . An isomorphism ψ : B1 → B2 of gln -crystals is an isomorphism of In -colored oriented graphs which preserves wt, εi , and ϕi (i ∈ In ). We say that B1 is isomorphic to B2 , and write B1 B2 . For bi ∈ Bi (i = 1, 2), let C(bi ) denote the connected component of bi as an In -colored oriented graph. We say that b1 is (gln -) crystal equivalent to b2 if there is an isomorphism of gln -crystals C(b1 ) → C(b2 ) sending b1 to b2 , and write b1 gln b2 . 3.2. Crystal of words For n ≥ 2, let Bn = { 1 < 2 < · · · < n } be a linearly ordered set. Then Bn can be made into a gln -crystal whose associated In -colored oriented graph is given by 1

2

n−2

n−1

1 −→ 2 −→ · · · −→ n − 1 −→ n, where wt(b) = b . Note that Bn is the crystal graph of the natural representation K(q)⊕n [26]. Let Wn be the set of all ﬁnite words with letters in Bn . That is, ⎛ ⎞ ⎠ ⊕ {∅}, B⊗r Wn = ⎝ n r≥1

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where ∅ denotes the empty word. Then Wn is a gln -crystal, where {∅} forms a trivial crystal, that is, wt(∅) = 0, ei ∅ = fi ∅ = 0, and εi (∅) = ϕi (∅) = 0 for all i ∈ In . In fact, Wn is the crystal graph of the tensor algebra generated by the natural representation. For simplicity, let us write a nonempty word w1 ⊗ · · · ⊗ wr ∈ B⊗r n as w1 · · · wr . Following the tensor product rule of the crystals, we can describe the Kashiwara operators ei , fi : Wn → Wn ∪ {0} (i ∈ In ) induced from those on Bn explicitly; (1) Suppose that a nonempty word w = w1 · · · wr is given. To each letter wk , we assign ⎧ ⎪ if wk = i, ⎨+, (i) (wk ) = −, if wk = i + 1, ⎪ ⎩ ·, otherwise, and let (i) (w) = ((i) (w1 ), . . . , (i) (wr )). (2) We replace a pair ((i) (ws ), (i) (ws )) = (+, −) such that s < s and (i) (wt ) = · for s < t < s with ( · , · ) in (i) (w), and repeat this process as long as possible until we get a sequence with no + placed to the left of −. This is called the i-signature of w. (3) We call the right-most − in the i-signature of w the i-good − sign, and deﬁne ei w to be the word obtained by applying ei to i + 1 corresponding to the i-good − sign. If there is no i-good − sign, then we deﬁne ei w = 0. (4) We call the left-most + in the i-signature of w the i-good + sign, and deﬁne fi w to be the word obtained by applying fi to i corresponding to the i-good + sign. If there is no i-good + sign, then we deﬁne fi w = 0. Example 3.3. Suppose that w = 1 2 4 2 1 1 2 3 2 1 1 3 ∈ B⊗12 4 . Then (1) (w) = ( + , − , · , − , + , + , − , · , − , + , + , · ) =(+

,−

, · , −, +

,+

,−

, · ,−

, +, +, ·) =( · , · , · , , · , · , · , · , · , ⊕, +, ·) where ⊕ and denote the i-good signs. We have e1 w = 1 2 4 1 1 1 2 3 2 1 1 3,

f1 w = 1 2 4 2 1 1 2 3 2 2 1 3.

3.3. Crystal of a Young tableau A partition is a nonincreasing sequence of nonnegative integers λ = (λk )k≥1 such that all but a ﬁnite number of its terms are zero. Each λk is called a part of λ,

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and the number of nonzero parts is called the length of λ and denoted by (λ). We m1 , 2m2 , . . .), where m is the number of occurrences of i in λ, and also write i λ = (1 |λ| = k≥1 λk = i≥1 imi . Recall that a partition λ = (λk )k≥1 is identiﬁed with a Young diagram, which is a collection of nodes (or boxes) in left-justiﬁed rows with λk nodes in the k-th row numbered from top to bottom. We denote by P the set of all partitions, and set Pn = { λ ∈ P | (λ) ≤ n } for n ≥ 1. A Young tableau T is obtained by ﬁlling a Young diagram λ with the entries from N. We say that T is semistandard if the entries in each row (resp. column) are weakly (resp. strictly) increasing from left to right (resp. from top to bottom). If all the entries in T are distinct, then T is called standard. We say that λ is the shape of T , and write sh(T ) = λ. For λ ∈ Pn , let Bn (λ) be the set of all semistandard (Young) tableaux of shape λ with entries in Bn = { 1, . . . , n }. For each T ∈ Bn (λ), let w(T ) (or wcol (T )) be the word obtained by reading the entries of T column by column from right to left, and in each column from top to bottom. Note that a semistandard tableau can be uniquely recovered from the associated word. Hence the map T → w(T ) gives a natural embedding of Bn (λ) into Wn , and we may view Bn (λ) as a subset of Wn . Example 3.4. B5 ((3, 2))

1 2 2 −→ 2 2 5 1 4 ∈ W5 4 5 Column reading

For T ∈ Bn (λ), the weight of T is given by wt(T ) = b∈Bn μb b ∈ Pn , where μb is the number of occurrences of b in T . Indeed, Bn (λ) together with 0 is stable under ei , fi (i ∈ In ), and hence Bn (λ) is a subcrystal of Wn . Theorem 3.5 ([26]). For λ = (λ1 , . . . , λn ) ∈ Pn , Bn (λ) is a gln -crystal, which is isomorphic to the crystal graph of the irreducible highest weight module with highest weight b∈Bn λb b ∈ Pn+ . Remark 3.6. (1) For λ ∈ Pn , let Hnλ be the semistandard tableaux in Bn (λ) such that each k-th row is ﬁlled with k (1 ≤ k ≤ n). Then wt(Hnλ ) ≥ wt(T ) for all T ∈ Bn (λ), that is, Hnλ is the highest weight vector of weight λ1 1 + · · · + λn n 4 . Hence, each T ∈ Bn (λ) can be obtained from Hnλ by applying a ﬁnite number of fi . (2) The way of reading the entries of a tableau by which we give a crystal structure on Bn (λ) is not unique. In fact, we may also deﬁne ei , fi on Bn (λ) with respect to various readings (see, for example, [2]).

4 We may identify a partition λ = (λ , . . . , λ ) with a dominant integral weight λ + · · · + λ . n n n 1 1 1

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Example 3.7. Here is the gl3 -crystal B3 ((2, 1)), where the action of the fi on semistandard tableaux is given by that on the associated words in W3 . 1 1 2 ↓1 1 2 2

2

1 3

2

1 3 2 ↓2 1 3 3

−→ −→

1

1 2 3 ↓1 2 2 3 ↓2 2 3 3

1

−→

1

−→

In general, we can also deﬁne a gln -crystal for semistandard tableaux of skew shape. For λ, μ ∈ P such that λ ⊃ μ (that is, λk ≥ μk for all k), let λ/μ be the diagram given by removing the nodes of μ from λ. Let Bn (λ/μ) be the set of all semistandard tableaux of shape λ/μ with entries in Bn . Just as mentioned earlier, we can check that Bn (λ/μ) is a gln -crystal. The crystal Bn (λ/μ) is not necessarily connected. We will discuss its explicit decomposition into its connected components in Section 4. 3.4. Crystal of a rational Young tableau Let us describe the crystal graph of the irreducible highest weight Uq (gln )-module with arbitrary highest weight λ = (λ1 , . . . , λn ) ∈ Zn+ = Pn+ . An element λ = (λ1 , . . . , λn ) in Zn+ is called a generalized partition of length n, and it may be identiﬁed with a generalized Young diagram in the following way. First, we ﬁx a vertical line. Then for each λk , we place |λk | nodes (or boxes) in the k-th row in a left-justiﬁed (resp. right-justiﬁed) way with respect to this vertical line if λk ≥ 0 (resp. λk ≤ 0). For example,

λ = (3, 2, 0, −1, −2)

−

←→

•

−2

− • •

−1

• • −

• • −

• −

1

2

3

← column number

where the horizontal line denotes an empty row. We enumerate the columns of a diagram as mentioned earlier. Let B∨ n = { −n < −n + 1 < · · · < −1} be the dual crystal of Bn whose associated graph is given by n−1

n−2

1

−n −→ −n + 1 −→ · · · −→ −1, where wt(−b) = −wt(b) = −b for b ∈ Bn (cf [24]). Let T be a tableau obtained by ﬁlling a generalized Young diagram λ = (λ1 , . . . , λn ) of length n with the entries in Bn B∨ n . We call T a rational semistandard tableau of shape λ if

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(1) the entries in the columns indexed by positive (resp. negative) numbers are from Bn (resp. B∨ n ), (2) the entries in each row (resp. column) are weakly (resp. strictly) increasing from left to right (resp. from top to bottom), (3) if b1 < · · · < bs (resp. −b1 < · · · < −bt ) are the entries in the 1-st (resp. the −1-st) column (s + t ≤ n), then bi ≤ bi for 1 ≤ i ≤ s, where {b1 < · · · < } = B \{b , · · · , b }. bn−t n t 1 We denote by Bn (λ) the set of all rational semistandard tableaux of shape λ. Remark 3.8. The notion of rational semistandard tableaux was introduced by Stembridge [42] to describe the characters of rational representations of gln over the complex numbers. Example 3.9. For λ = (3, 2, 0, −1, −2), we have

− −4

− −5 −3

1 3 −

2 4 −

2 − ∈ B5 (λ).

⊗N (where N = Now, embed Bn (λ) into (Bn ⊕ B∨ n) k≥1 |λk |) by column reading of tableaux, and apply ei , fi to Bn (λ) (i ∈ In ). Then Bn (λ) together with 0 is stable under ei , fi , and becomes a gln -crystal, which is in fact isomorphic to the crystal graph of the irreducible highest weight Uq (gln )-module with highest weight λ (which is b∈Bn λb b ). Let us explain the relation between the crystals of rational semistandard tableaux and ordinary semistandard tableaux. Let T be a rational semistandard tableau in Bn ((0n−t , −1t )) (0 ≤ t ≤ n) with the entries −b1 < · · · < −bt in the −1-st column. in We deﬁne σ(T ) be the tableau in Bn ((1n−t , 0t )) with the entries b1 < · · · < bn−t the 1-st column, where {b1 < · · · < bn−t } = Bn \ {b1 < · · · < bt }. If t = n, then we deﬁne σ(T ) to be the empty tableau. In general, for λ ∈ Zn+ and T ∈ Bn (λ), we deﬁne σ(T ) to be the tableau obtained by applying σ to the −1-st column of T . Note that the shape of σ(T ) is λ + (1n ). For example, when n = 5, we have ⎞ ⎛ 1 2 2 1 1 2 2 ⎟ ⎜ 3 4 2 3 4 ⎟ ⎜ ⎟ = − − − 4 − − σ⎜ . ⎟ ⎜ ⎠ ⎝ −5 − − − − − −4 −3 −4 Lemma 3.10. For λ ∈ Zn+ , the map σ : Bn (λ) ∪ {0} → Bn (λ + (1n )) ∪ {0},

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J.-H. Kwon

(σ(0) = 0) is a bijection which commutes with ei , fi for i ∈ In , where wt(σ(T )) = wt(T ) + (1 + · · · + n ) for T ∈ Bn (λ). In particular, Bn (λ) and Bn (λ + (1n )) gives the same In -colored oriented graph. In other words, the crystal Bn (λ) for λ ∈ Zn+ is essentially the same as the crystal of ordinary semistandard tableaux Bn (λ + (d n )) for λ + (d n ) ∈ Pn . In particular, it is connected with the unique highest weight element, say Hnλ . Example 3.11. For λ = (3, 2, 0, −1, −2),

Hnλ =

− −5

− −5 −4

1 2 −

1 2 −

1 −

λ+(25 ) is the unique highest weight element in B5 (λ). Note Hnλ = σ −2 Hn . 4. Crystal equivalence In classical representation theory, the decompositions of ﬁnite-dimensional gln modules are explained using the characters of the representations, which are symmetric polynomials in n variables. In particular, there are well-known combinatorial algorithms, called the Robinson–Schensted correspondence and the Littlewood–Richardson rule, which provide in a bijective way the character identities corresponding to the decomposition of the tensor power of the natural representation and the tensor product of two irreducible highest weight representations, respectively (for a background on the combinatorics of Young tableaux, which will be needed in this section, see [10, 37, 41]). In this section, we will see that these algorithms obtain more concrete representation theoretical meanings in the language of crystals, that is, they yield morphisms of crystals, and hence explain the decompositions of the corresponding crystals directly. 4.1. Knuth equivalence For a given word w ∈ Wn and three consecutive letters xyz in w, consider the transformations w = . . . xyz . . . −→ w = . . . xzy . . . , w = . . . xyz . . . −→ w = . . . yxz . . . ,

if y ≤ x < z, if y < x ≤ z,

which are called the elementary Knuth transformations. Then, we say that two words w and w in Wn are Knuth equivalent if they can be transformed into each other by a sequence of elementary Knuth transformations and their inverses, and write w K w .

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In general, for two semistandard tableau T and T with entries in { 1, . . . , n }, we deﬁne T K T if w(T ) K w(T ). Remark 4.1. Note that the following two semistandard tableaux (of skew shapes) are Knuth equivalent. ⎧ x ⎪ ⎪ ⎪ if y ≤ x < z, ⎨ y z, y x K z ⎪ y ⎪ ⎪ , if y < x ≤ z ⎩ z x where the tableau on the right-hand side can be obtained by sliding the two entries of the tableau in the left-hand side in such a way that it still remains semistandard, and also vice versa. Next, let us recall the Schensted column bumping algorithm for semistandard tableaux: For λ ∈ Pn and T ∈ Bn (λ), we deﬁne (T ← b) (b ∈ Bn ) to be the tableau obtained from T by applying the following procedure; (1) Let b be the smallest entry in the ﬁrst (or the left-most) column that is greater than or equal to b. (2) Replace b by b (b is bumped out of the ﬁrst column). If there is no such b , put b at the bottom of the ﬁrst column and stop the procedure. (3) Repeat (1) and (2) on the next column with b . Note that (T ← b) ∈ Bn (μ) for some μ ∈ Pn , where μ is obtained by adding a node to λ. Now, for a given word w = w1 · · · wr ∈ Wn , deﬁne P(w) = ((· · · ((w1 ← w2 ) ← w3 ) · · · ) ← wr ). P(w) is called the P-tableau of w or the insertion tableau of w. Because the Knuth equivalence of words with three letters can be understood in terms of sliding of entries in skew diagrams (see Remark 4.1), it is not difﬁcult to see that w K P(w) (or w K w(P(w))). The following is the key fact, which connects the theory of crystals and the classical combinatorics of Young tableaux; Theorem 4.2 ([35]). For w, w ∈ Wn , we have w K w ⇐⇒ w gln w . Proof. (⇒) By deﬁnition of the tensor product of crystals, we may assume that w = xyz and w is transformed to w by an elementary Knuth transformation. Identifying w and w with the column words of the corresponding semistandard tableaux of skew shapes respectively (see Remark 4.1), it is straightforward to check that the elementary Knuth transformation naturally extends to an isomorphism of gln -crystals between the connected components of two words w and w .

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J.-H. Kwon

(⇐) Since w K P(w) and w K P(w ), we have P(w) gln P(w ). Suppose that sh(P(w)) = λ and sh(P(w )) = μ. There exists a sequence of indices i1 , · · · , ir ∈ In and nonnegative integers ak (1 ≤ k ≤ r) such that eia11 . . . eiarr P(w) = Hnλ . Since P(w) gln P(w ), we must have eia11 . . . eiarr P(w ) = Hnμ . μ

μ

Hence, we get wt(Hnλ ) = wt(Hn ), which implies λ = μ and Hnλ = Hn . Since P(w) = fiar r · · · fia11 Hnλ , we conclude that P(w) = P(w ). In particular, we have w K w Corollary 4.3. Each connected component in Wn is isomorphic to a Bn (λ) for some λ ∈ Pn . Corollary 4.4 (cf [30]). Let T and T be two semistandard tableaux. If T K T , then T = T . That is, for a given word w ∈ Wn , there exists a unique semistandard tableau T which is Knuth equivalent to w.

4.2. Robinson–Schensted correspondence Let us consider the decomposition of B⊗r n (r ≥ 2). For w = w1 . . . wr ∈ Wn , suppose that P(w) is of shape λ. Then we deﬁne Q(w) to be the standard tableau of shape λ such that if wi is inserted into (((· · · ( w1 ← w2 ) · · · ) ← wi−1 ) to create a node in λ, then we ﬁll the node with i. We call Q(w) the Q-tableau of w or the recording tableau. Example 4.5. For example, if w = 31224 ∈ W4 , then we have 1 P(w) = 2 4

2

3 ,

1 Q(w) = 3 5

2

4 .

Then the Robinson–Schensted correspondence is that the map w → (P(w), Q(w)) ⊗r for

w ∈ Wn gives a one-to-one correspondence between the two sets, Bn and λ Bn (λ) × STr (λ), where the union is taken over all partitions λ ∈ Pn such that |λ| = r, and STr (λ) denotes the set of all standard tableaux of shape λ with entries in { 1, . . . , r }. Because each connected component in B⊗r n is isomorphic to a Bn (λ) for some λ ∈ Pn with |λ| = r and Bn (λ) is connected, the multiplicity of Bn (λ) in B⊗r n , say mλ , λ , that is, m = |ST (λ)| such that P(w) = H is equal to the number of the words in B⊗r λ r n n by the Robinson–Schensted correspondence.

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Moreover, we can check that for w ∈ B⊗r n and i ∈ In , Q(w) = Q(fi w),

if fi w = 0,

(equivalently, Q(w) = Q(ei w), if ei w = 0). In other words, the Q-tableaux are invariant under the Kashiwara operators, and it turns out that the Q-tableau of w represents the connected component of w in B⊗r n . Summarizing the aforementioned arguments, we have the following crystal version of the Robinson–Schensted correspondence; Theorem 4.6 ([31]). The map w → (P(w), Q(w)) gives the following isomorphism of gln -crystals; B⊗r Bn (Q), n λ∈Pn ,|λ|=r Q∈STr (λ)

where Bn (Q) = Bn (λ) × { Q } for λ ∈ Pn with |λ| = r and Q ∈ STr (λ) is a gln -crystal isomorphic to Bn (λ). A word w = w1 . . . wr ∈ Wn is called a lattice permutation if for 1 ≤ k ≤ r and 1 ≤ i ≤ n − 1, the number of occurrences of i in w1 . . . wk is no less than that of i + 1. Then it follows from the crystal structure on Wn (see 3.2) that ei w = 0 for all i ∈ In (that is, P(w) = Hnλ for some λ ∈ Pn ) if and only if w is a lattice permutation. Remark 4.7. A connection between the Robinson–Schensted correspondence and the representations of Uq (gln ) was ﬁrst observed by Date et al. [6]. 4.3. Littlewood–Richardson rule Next, let us discuss the decomposition of Bn (μ) ⊗ Bn (ν) for μ, ν ∈ Pn . For T ∈ Bn (μ) and T ∈ Bn (ν), let w(T ) = w1 w2 . . . wr . We deﬁne P(T ⊗ T ) = ((((T ← w1 ) ← w2 ) · · · ) ← wr ) (usually denoted as (T ← T )). Clearly, we have T ⊗ T gln P(T ⊗ T ). Next, deﬁne Q(T ⊗T ) to be the semistandard tableau of shape λ/μ (λ = sh(P(T ⊗ T )) such that if wi is in the k-th row of T and inserted into (((· · · ( T ← w1 ) ← w2 ) · · · ) ← wi−1 ) to create a node in λ/μ, then we ﬁll the node with k. We call Q(T ⊗ T ) the recording tableau of (T ← T ). Example 4.8. Suppose that we have T =

1 3

2

2

2 and T = 3 4

2 4 .

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Then we have 1 2 P(T ⊗ T ) = 3 4

2 3 4

2

2 and

• • Q(T ⊗ T ) = 2 3

• • 1 2

1 .

Recall that given λ, μ, and ν in P such that μ ⊂ λ and |λ| = |μ| + |ν|, a semistandard tableau T of shape λ/μ is called a Littlewood–Richardson tableau of shape λ/μ with content ν if (1) the number of occurrences of k in T is equal to νk for k ≥ 1, (2) w(T) is a lattice permutation. We denote by LRμλ ν the set of all Littlewood–Richardson tableaux of shape λ/μ with content ν. Then, the map T ⊗ T → (P(T

⊗ T ), Q(T ⊗ T λ)) gives a bijection between the two sets Bn (μ) ⊗ Bn (ν) and λ∈Pn Bn (λ) × LR μ ν [44]. Because the connected components are completely determined by T ⊗ T such that P(T ⊗ T ) = Hnλ , the multiplicity of Bn (λ) in Bn (μ) ⊗ Bn (ν) is equal to |LR λμ ν | called the Littlewood– Richardson coefﬁcient. Moreover, as in the case of the Robinson–Schensted correspondence, we have for T ⊗ T ∈ Bn (μ) ⊗ Bn (ν) and i ∈ In , Q(T ⊗ T ) = Q(fi (T ⊗ T )),

if fi (T ⊗ T ) = 0.

Hence, the Q-tableau is invariant under the Kashiwara operators, and it represents the connected component of T ⊗ T in Bn (μ) ⊗ Bn (ν). Summarizing, we also have the crystal version of the Littlewood–Richardson rule; Theorem 4.9 (cf [40]). The map T ⊗ T → (P(T ⊗ T ), Q(T ⊗ T )) gives the following isomorphism of gln -crystals; Bn (μ) ⊗ Bn (ν) Bn (Q), λ∈Pn Q∈LR λμν

where Bn (Q) = Bn (λ)×{ Q } for λ ∈ Pn and Q ∈ LRλμ ν is a gln -crystal isomorphic to Bn (λ). For a skew Young diagram λ/μ, let us consider Bn (λ/μ). Then the decomposition of Bn (λ/μ) into its connected components reduces to ﬁnding all T ∈ Bn (λ/μ) equivalent to Hnν , where ν is the content (or weight) of T . For T ∈ Bn (λ/μ), it follows that T gln Hnν ⇐⇒ ei T = 0 for all i ∈ In ⇐⇒ w(T) is a lattice permutation ⇐⇒ T ∈ LR λμν .

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Therefore, we obtain the skew Littlewood–Richardson rule; Bn (λ/μ) Bn (T), ν∈Pn T ∈LR λμν

where Bn (T) is the connected component of T in Bn (λ/μ), which is isomorphic to Bn (ν). Remark 4.10. (1) There are analogues of the material in this section for classical Lie algebras of B, C, D type, using the Kashiwara and Nakashima description [1,32,33]. See also [2, 19] for a generalization to Lie superalgebras. (2) The combinatorics of Young tableaux presented here can be formalized using the notion of the plactic algebra introduced by Lascoux and Schützenberger [30], which is a noncommutative ring of the Knuth equivalence classes of the words in Wn with the multiplication given by juxtaposition (or tensor product) of words. In [35], using his path models, Littelmann gave a nice generalization of the plactic algebra for complex semisimple Lie algebras, which consists of the crystal equivalence classes of certain piecewise linear paths in the weight lattice, where the multiplication is given by the concatenation of paths.

5. Bicrystals Generalizing the Robinson–Schensted correspondence, Knuth introduced an algorithm that establishes a bijection between m × n matrices of nonnegative integers and pairs of semistandard tableaux of the same shape [27] (now called the Robinson– Schensted–Knuth correspondence). The associated character identity explains the decomposition of the symmetric algebra generated by the tensor product of the natural representations of glm and gln into irreducible representations of the Lie algebra glm ⊕ gln over the complex numbers [13]. In this section, we explain the Knuth result in terms of crystals (see [4, 29]), and discuss its applications to combinatorics and representation theory. 5.1. Robinson–Schensted–Knuth correspondence First, let us consider a crystal of biwords. Let m, n be nonnegative integers not both zero. Set

m,n = { (i, j) ∈ Wm × Wn | (1) i = i1 . . . ir and j = j1 . . . jr

for some r ≥ 0,

(2) (i1 , j1 ) ≤ · · · ≤ (ir , jr )}, where the ordering is a rather weird kind of lexicographic type ordering given by (i, j) < (k, l)

⇐⇒

(j < l) or (j = l and i > k)

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for (i, j) and (k, l) ∈ Bm × Bn . Now, for i ∈ Im and (i, j) ∈ m,n , we deﬁne ei (i, j) = (ei i, j),

fi (i, j) = (fi i, j),

where we assume that xi (i, j) = 0 if xi i = 0 (x = e, f ). We set wt(i, j) = wt(i) ∈ Pm , εi (i, j) = εi (i) and ϕi (i, j) = ϕi (i) (i ∈ Im ). Then the set of biwords m,n together with wt, ei , fi , εi , ϕi (i ∈ Im ) becomes a glm -crystal. In fact, it is isomorphic to Bm ( 1 ) ⊗ · · · ⊗ Bm ( n ),

1 ,..., n ∈Z≥0

where Bm ( ) is a glm -crystal with highest weight 1 for ≥ 0 (note that Bm ( j ) corresponds to subwords is is+1 . . . it of i in (i, j) ∈ m,n such that js = js+1 = . . . = jt = j). Next, set

∗m,n = { (k, l) ∈ Wm × Wn | (l, k) ∈ n,m }. Similarly, for j ∈ In and (k, l) ∈ ∗m,n , we deﬁne ej∗ (k, l) = (k, ej l),

fj∗ (k, l) = (k, fj l),

and set wt ∗ (k, l) = wt(l) ∈ Pn , ε∗j (k, l) = εj (l), and ϕj∗ (k, l) = ϕj (l) (j ∈ In ). Then also the set of biwords ∗m,n together with wt ∗ , ej∗ , fj∗ , ε∗j , ϕj∗ (j ∈ In ) is a gln -crystal, which is isomorphic to Bn ( 1 ) ⊗ · · · ⊗ Bn ( m ). ( 1 ,..., m )∈Zm ≥0

Let Mm×n be the set of all m × n matrices with nonnegative integral entries. We assume that the row (resp. column) numbers are indexed by Bm (resp. Bn ). For (i, j) ∈ m,n , deﬁne A(i, j) = (aij ) to be the matrix in Mm×n , where aij is the number of k’ such that (ik , jk ) = (i, j) for i ∈ Bm and j ∈ Bn . For (k, l) ∈ ∗m,n , we deﬁne A(k, l) ∈ Mm×n in the same way. Then the map (i, j) → A(i, j) (resp. (k, l) → A(k, l)) gives a bijection from m,n (resp. ∗m,n ) to Mm×n , where the pair of empty words (∅, ∅) corresponds to the zero matrix. With these identiﬁcations, Mm×n becomes a glm and gln -crystal, simultaneously. Example 5.1. Suppose that ⎛ 1 0 1 ⎜ 0 2 0 A=⎜ ⎝ 1 0 0 0 0 2

⎞ 1 0 ⎟ ⎟ 1 ⎠ ∈ M4×4 . 0

Then A = A(i, j) = A(k, l) for (i, j) ∈ 4,4 and (k, l) ∈ ∗4,4 , where i = 3 1 2 2 4 4 1 3 1, j = 1 1 2 2 3 3 3 4 4,

k = 1 1 1 2 2 3 3 4 4, l = 4 3 1 2 2 4 1 3 3.

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Deﬁnition 5.2. Suppose that B is a glm -crystal and also a gln -crystal under ei , fi (i ∈ Im ) and ej∗ , fj∗ (j ∈ In ), respectively. We call B a (glm , gln )-bicrystal if ei , fi commute with ej∗ , fj∗ (i ∈ Im , j ∈ In ), where we understand the Kashiwara operators as the associated maps from B ∪ {0} to itself (that is, xi 0 = xj∗ 0 = 0, for x = e, f ). For a (glm , gln )-bicrystal B, if we put Im,n = Im In∗ (In∗ = { k∗ | 1 ≤ k ≤ n − 1 }), then B can be viewed as an Im,n -colored oriented graph, where j∗

b → b

if and only if b = fj∗ b

(j ∈ In ).

Example 5.3. For λ ∈ Pm and μ ∈ Pn , Bm (λ) × Bn (μ) is a (glm , gln )-bicrystal where xi (b1 , b2 ) = (xi b1 , b2 ), xj∗ (b1 , b2 ) = (b1 , xj∗ b2 ) for (b1 , b2 ) ∈ Bm (λ)×Bn (μ) and x = e, f . Now, we can also check that Mm×n is a (glm , gln )-bicrystal. For A ∈ Mm×n , suppose that A = A(i, j) = A(k, l) for (i, j) ∈ m,n and (k, l) ∈ ∗m,n . Let us deﬁne P(A) = P(i),

Q(A) = P(l).

Note that P(A) glm A and Q(A) gln A by deﬁnition of the crystal structures on Mm×n . Since Mm×n is a (glm , gln )-bicrystal, it follows that P(fj∗ A) = P(A),

if fj∗ A = 0 for j ∈ In ,

Q(fi A) = Q(A),

if fi A = 0 for i ∈ Im .

In other words, fj∗ A glm A and fi A gln A. Let C be a connected component in Mm×n as an Im,n -colored oriented graph. For A ∈ C, suppose that P(A) ∈ Bm (λ) and Q(A) ∈ Bn (μ) for some λ ∈ Pm and μ ∈ Pn . Then the map A → (P(A), Q(A)) extends to an isomorphism of (glm , gln )-bicrystals; C −→ Bm (λ) × Bn (μ) (for detailed veriﬁcations, see [4, 28]). Hence, to describe the decomposition of Mm×n as a (glm , gln )-bicrystal, it sufﬁces to ﬁnd all the highest weight elements A in Mm×n , that is, those A for μ which (P(A), Q(A)) = Hmλ , Hn for some λ ∈ Pm and μ ∈ Pn , or equivalently, ei A = ej∗ A = 0 for all i ∈ Im and j ∈ In . It is not difﬁcult to verify that A = (aij ) is of highest weight if and only if (1) aij = 0 unless i = j, and (2) the diagonal entries, say λi = aii for 1 ≤ i ≤ min{m, n}, are weakly decreasing, and hence they form a partition λ = (λi )i≥1 ∈ Pm ∩ Pn . For example, if m ≤ n, then ⎞ ⎛ λ1 ⎟ ⎜ λ2 ⎟ ⎜ ⎟ ⎜ .. A=⎜ ⎟. . ⎟ ⎜ ⎠ ⎝ λm−1 λm

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Then, the connected component of A is isomorphic to Bm (λ) × Bn (λ). Therefore, we obtain the crystal version of the Robinson–Schensted–Knuth correspondence. Theorem 5.4 ([4]). The map A → (P(A), Q(A)) gives the following isomorphism of (glm , gln )-bicrystals; Mm×n Bm (λ) × Bn (λ). λ∈Pm ∩Pn

Remark 5.5. The map A → (P(A), Q(A)) above is slightly different from Knuth’s original description. But if we give Mm×n another bicrystal structure by replacing Bm and Bn (the index sets of rows and columns in Mm×n ) with their duals, then we recover the original Knuth map with a little modiﬁcation, which amounts to a morphism of crystals. (See [10, 28] for other variations of the Knuth correspondence and bicrystal structures on Mm×n .) As an application to combinatorics, let us consider the crystal of symmetric matrices with respect to a diagonal action of the Kashiwara operators. For n ≥ 2, set Mn = { A ∈ Mn×n | A = At }. For i ∈ In and A ∈ Mn , we deﬁne ei A = ei ei∗ A = ei∗ ei A,

fi A = fi fi∗ A = fi∗ fi A.

Then we have (ei A)t = (ei ei∗ A)t = ei∗ ei At = ei A ∈ Mn ∪ {0}, (fi A)t = (fi fi∗ A)t = fi∗ fi At = fi A ∈ Mn ∪ {0}. Hence, ei , fi : Mn → Mn ∪ {0} are well-deﬁned operators. Note that P(A) = Q(A), and P(ei A) = ei P(A),

P(fi A) = fi P(A).

Now, if we put wt(A) = wt(P(A)) = wt(Q(A)) ∈ Pn , εi (A) = max{ k : eik A = 0 }, and ϕi (A) = max{ k : fik A = 0 } for i ∈ In and A ∈ Mn , then Mn is a gln -crystal. Because the highest weight elements in Mn are exactly those in Mn×n , we have Proposition 5.6 (cf [27]). Mn is a gln -crystal, which decomposes as follows; Mn Bn (λ). λ∈Pn

For A = (aij ) ∈ Mn , let o(A) = | { i ∈ Bn | aii ≡ 1 (mod 2) } | . With the help of the crystal structure on Mn , we may easily deduce the following fact, which comes from [27].

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Proposition 5.7. Let Mn (k) = { A ∈ Mn | o(A) = k } for k ≥ 0. Then Mn (k) is a subcrystal of Mn , and decomposes as follows; Bn (λ), Mn (k) λ∈Pn o(λ)=k

where o(λ) is the number of odd parts in λ. Proof. Note that for A ∈ Mn and i ∈ In , if fi A = 0, then tr(fi A) = tr(A) or tr(A) ± 2. This implies that Mn (k) together with 0 is stable under ei and fi , and hence it is a subcrystal of Mn . Moreover, if A is a highest weight element in Mn (k), that is, P(A) = Hnλ for some λ = (λ1 , . . . , λn ) ∈ Pn , then A is a diagonal matrix with diagonal entries λ1 , . . . , λn , and o(A) = k is equal to the number of odd parts in λ. In particular, we have Mn (0) Bn (λ), λ∈Pn λ : even

where a partition λ is called even if each part of λ is an even integer. Remark 5.8. A relation between the diagonal entries of a symmetric matrix and the shape of the corresponding tableau was ﬁrst observed by Knuth [27]. The present exposition is a variation. (See [28] for a general statement including the result in [27]). 5.2. Dual Robinson–Schensted–Knuth correspondence Let us now discuss the dual Robinson–Schensted–Knuth correspondence. Let Mm×n be the set of all m × n matrices with entries 0 or 1. We also assume that the row (resp. column) numbers are indexed by Bm (resp. Bn ). As in the case of Mm×n , we identify a matrix in Mm×n with a biword reading the row and column indices of nonzero entries in the matrix with respect to a certain linear ordering. First, we set

m,n = { (i, j) ∈ Wm × Wn | (1) i = i1 . . . ir and j = j1 . . . jr for some r ≥ 0, (2) (i1 , j1 ) ≺ · · · ≺ (ir , jr ) }, where for (i, j) and (k, l) ∈ Bm × Bn , the linear ordering ≺ (right to left lexicographic ordering) is given by (i, j) ≺ (k, l)

⇐⇒

(j < l) or (j = l and i < k).

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J.-H. Kwon

We deﬁne ei , fi : m,n → m,n ∪ {0} (i ∈ Im ) by ei (i, j) = (ei i, j),

fi (i, j) = (fi i, j),

for (i, j) ∈ m,n . Set wt(i, j) = wt(i) ∈ Pm , εi (i, j) = εi (i), and ϕi (i, j) = ϕi (i) (i ∈ Im ). Then m,n is a glm -crystal, which is isomorphic to Bm (1 1 ) ⊗ · · · ⊗ Bm (1 n ), 0≤ 1 ,..., n ≤m

where Bm (1 ) is a glm -crystal with highest weight 1 + · · · + for 0 ≤ ≤ m. Next, we set ∗m,n = { (k, l) ∈ Wm × Wn | (1) k = k1 · · · kr and l = l1 · · · lr for some r ≥ 0, (2) (k1 , l1 ) ≺ · · · ≺ (kr , lr ) }, where for (i, j) and (k, l) ∈ Bm × Bn , the linear ordering ≺ is given by (i, j) ≺ (k, l)

⇐⇒

(i > k) or (i = k and j < l).

Similarly, we deﬁne ej∗ , fj∗ : ∗m,n → ∗m,n ∪ {0} (j ∈ In ) by ej∗ (k, l) = (k, ej l),

fj∗ (k, l) = (k, fj l),

for (k, l) ∈ ∗m,n . Set wt ∗ (k, l) = wt(l) ∈ Pn , ε∗j (k, l) = εj (l), and ϕj∗ (k, l) = ϕj (l) (j ∈ In ). Then ∗m,n is a gln -crystal isomorphic to Bn (1 1 ) ⊗ · · · ⊗ Bn (1 m ). 0≤ 1 ,..., m ≤n For (i, j) ∈ m,n (resp. (k, l) ∈ ∗m,n ), we deﬁne A(i, j) ∈ Mm×n (resp. A(k, l) ∈ Mm×n ) in the same way as in 5.1, which yields a bijection between m,n (resp. ∗m,n ) . Under these maps, Mm×n becomes a (glm , gln )-bicrystal. For A ∈ Mm×n , and Mm×n

if we deﬁne P(A) = P(i) and Q(A) = P(l), where A = A(i, j) = A(k, l) for (i, j) ∈ m,n and (k, l) ∈ ∗m,n , then the map A → (P(A), Q(A)) gives an isomorphism of to Bm (λ) × Bn (μ) (glm , gln )-bicrystals from each connected component in Mm×n for some λ ∈ Pm and μ ∈ Pn . is a highest weight element if and only Now, we can check that A = (aij ) ∈ Mm×n if it satisﬁes the following condition; aij = 0 =⇒ ai+1 j = ai j+1 = 0, whenever i + 1 ∈ Bm and j + 1 ∈ Bn . Example 5.9. For m = 4 and n = 8, ⎛ 1 1 1 1 1 1 1 ⎜1 1 1 1 1 1 A=⎜ ⎝1 1 1 1 1

⎞ 1 ⎟ ⎟ ⎠

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497

is a highest weight element, and (P(A), Q(A)) = (Hmλ , Hnλ ), where λ = (8, 6, 4, 1) and λ is the conjugate of λ (cf [37]). In general, a highest weight element A = (aij ) ∈ Mm×n , put λ = (λ1 , . . . , λm ) for n a for 1 ≤ i ≤ m. Then λ is a partition and (P(A), Q(A)) = with λi = ij j=1

(Hmλ , Hnλ ). Summarizing the aforementioned arguments, we have

Theorem 5.10 ([4]). The map A → (P(A), Q(A)) gives the following isomorphism of (glm , gln )-bicrystals; Mm×n Bm (λ) × Bn (λ ). λ∈Pm λ ∈Pn

is not obtained from its glm -crystal Remark 5.11. The gln -crystal structure on Mm×n structure just by taking the transpose of a matrix, as in case of Mm×n . The reasons for this can be found in [28].

5.3. Decomposition of Fock space representations Let gl∞ be the Lie algebra of Z × Z-matrices over K with only ﬁnite number of nonzero entries. As an application of the dual Robinson–Schensted–Knuth correspondence, let us give an explicit decomposition of the n-fold tensor power of the Fock space representation of gl∞ or Uq (gl∞ ). The quantum groupUq (gl∞ ) is deﬁned in a similar way as Uq (gln ), where the weight lattice is P = i∈Z Zi , the simple root is αi = i − i+1 , the simple coroot hi satisﬁes hi , λ = (αi , λ) for all λ ∈ P (i ∈ Z), and so on. Let us denote by I the index set for the simple roots. To consider integrable highest weight Uq (gl∞ )-modules, we will use the extended weight lattice P instead of P, where P = Z0 ⊕ Zi ,

i∈Z

and 0 ∈ Hom i∈I Zhi , Z is given by hi , 0 = δ0i for i ∈ I. For c ∈ Z, let c be the fundamental weight, that is, c (hi ) = δic for i ∈ I. Then all the results in Section 2 hold for Uq (gl∞ ), and we can thus deﬁne the notion of (abstract) gl∞ -crystals in the same way. Now, consider the following gl∞ -crystal B; −2

−1

0

1

2

· · · −→ −1 −→ 0 −→ 1 −→ 2 −→ · · ·, where wt(b) = b for b ∈ Z. Note that B is the crystal graph of the natural representation of Uq (gl∞ ). Unlike gln -crystals, a connected component in B⊗r does not contain a highest weight element. So, we use the q-deformed Fermionic Fock space (or semi-inﬁnite wedge space) representation F of Uq (gl∞ ) [11] and its crystal graph (cf [38]) to describe the crystal graphs of integrable highest weight Uq (gl∞ )-modules.

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Let F be the set of semi-inﬁnite words w = · · · w−3 w−2 w−1 w0 with letters in B such that (1) wi−1 < wi for i ≤ 0, (2) there exists an integer c ∈ Z such that wi = i + c for i

0,

where c is called a charge of w. For example, the charge of w = · · · − 5− 4− 1 2 4 ∈ F is −1. For w ∈ F, we deﬁne wt(w) = 0 + mi i ∈ P, i∈B

where mi = { k | wk = i } − δ−i,|i| . Because mi = 0 for almost all i ∈ B, wt(w) is well-deﬁned. For each i ∈ I, we deﬁne the Kashiwara operators ei , fi : F −→ F ∪ {0} as in the case of Wn (see 3.2). Since for w = · · · w−3 w−2 w−1 w0 ∈ F and i ∈ I, (i) (w) = (· · · , (i) (w−3 ), (i) (w−2 ), (i) (w−1 ), (i) (w0 )) has at most one + or − (we read the signs from left to right), ei and fi (i ∈ I) are well-deﬁned. Set εi (w) = max{ k : eik w = 0 } and ϕi (w) = max{ k : fik w = 0 }. Then F becomes a gl∞ -crystal, and it is isomorphic to the crystal graph of the Fock space representation F (cf [38]). Remark 5.12. Note that a semi-inﬁnite word w in F of charge c can be identiﬁed with a partition λ = (λk )k≥1 such that λk = w1−k − c + k − 1 for k ≥ 1. For example, w = · · · − 5− 4− 1 1 4

←→

• λ= • •

• • •

• •

•

•

.

Let B(c) be the set of words in F of charge c ∈ Z. Then B(c) is a subcrystal of F isomorphic to the crystal graph of the irreducible highest weight module with highest weight c (cf [38]). Hence, we have B(c), F= c∈Z

where B(c) is a connected component with the highest weight element H c = · · · c − 3 c − 2 c − 1 c. Let λ = (λ1 , . . . , λn ) ∈ Zn+ be a generalized partition of length n. We call an n-tuple of semi-inﬁnite words w = (w(1) , · · · , w(n) ) a semi-inﬁnite semistandard tableau of shape λ if (i)

(i)

(i)

(1) w(i) = · · · w−2 w−1 w0 ∈ B(λn−i+1 ) for 1 ≤ i ≤ n, (i)

(i+1)

(2) wk ≥ wk+di for 1 ≤ i < n and k ≤ 0, where di = λn−i+1 − λn−i .

We denote by B(λ) the set of all semi-inﬁnite semistandard tableaux of shape λ. We may identify each w ∈ B(λ) with a semi-inﬁnite tableau with inﬁnitely many rows and n columns, where each row of w reads (from left to right) as follows; (n)

(3)

(2)

(1)

wk+d1 +···+dn−1 · · · wk+d1 +d2 wk+d1 wk

(k ≤ 0).

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499

Example 5.13. Let λ = (3, 1, −2, −3), and let w = (w(1) , w(2) , w(3) , w(4) ) be given by w(4) .. . −4 −3 −2 −1 0 2 3 5

w(3) .. . −4 −3 −2 −1 1 3

w(2) .. . −4 −2 0

w(1) .. . −4 −1

.

Then w is a semi-inﬁnite semistandard tableau of shape λ. For λ = (λ1 , . . . , λn ) ∈ Zn+ and w = (w(1) , . . . , w(n) ) ∈ B(λ) , we may assume that w = w(1) ⊗ · · · ⊗ w(n) ∈ F⊗n , and consider B(λ) as a subset of F⊗n . Then we can check that B(λ) together with 0 is stable under ei and fi (i ∈ I). Hence, B(λ) is a gl∞ -crystal. Moreover, it is connected with the unique highest weight element H λ = H λn ⊗ · · · ⊗ H λ1 . Theorem 5.14. For λ = (λ1 , . . . , λn ) ∈ Zn+ , B(λ) is a gl∞ -crystal, which is isomorphic to the crystal graph of the irreducible highest weight module with highest weight λ1 + · · · + λn . Remark 5.15. Since each word in F can be identiﬁed with a partition, a semi-inﬁnite semistandard tableau in B(λ) can be identiﬁed with an n-tuple of partitions where the semistandard condition can be expressed in terms of certain inclusions among the associated partitions. This identiﬁcation naturally gives rise to a realization of the crystal graph of an irreducible highest weight representation of an afﬁne quantum group (cf [8, 16, 38]). Here we will use the tableaux realization of B(λ) to apply the combinatorics of Young tableaux given in the previous sections. Our goal is to give an explicit decomposition of F⊗n for n ≥ 2. We start with another description of F⊗n in terms of matrices of nonnegative integers. Set = {A = (aij )i∈B,j∈Bn | M∞×n (1) aij ∈ { 0, 1 }, (2) aij = 1 for all i

0, and aij = 0 for all i " 0 }.

For w = w(1) ⊗ · · · ⊗ w(n) ∈ F⊗n , set A(w) = (aij ) ∈ M∞×n , where aij is the (j) number of occurrences of i in w (1 ≤ j ≤ n). The map w → A(w) is a bijection , where each w(j) corresponds to the j-th column of A(w) for from F⊗n to M∞×n 1 ≤ j ≤ n.

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Example 5.16.

F⊗3

⎛

.. .. .. . . . −4 −4 −4 −3 −3 −1 w = −2 ⊗ −1 ⊗ 0 2 3 1 3 2 3

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ −→ A(w) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

.. . 1 1 1 0 0 0 1 1 0 .. .

.. . 1 1 0 1 0 0 0 1 0 .. .

.. . 1 0 0 1 1 1 1 1 0 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∈ M ∞×3 , ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where the horizontal line is between the 0-th row and the 1-st row of the matrix. Now deﬁne a bicrystal structure on M∞×n , which is naturally induced from the one on Mm×n in 5.2. For m > 0, we deﬁne Resm : M∞×n −→ M2m+1×n by Resm A = (aij )−m≤i≤m, j∈In for A = (aij ) ∈ M∞×n , where we assume that instead of B2m+1 and I2m+1 , the row index set for M2m+1×n and the index set for the simple roots of gl2m+1 are given by { −m, . . . , −1, 0, 1, . . . , m } and {−m + 1, . . . , and i ∈ I, we deﬁne ei A and fi A to be 0, . . . , m − 1}, respectively. For A ∈ M∞×n the unique elements in M∞×n ∪ {0} satisfying

Resm (ei A) = ei (Resm A),

Resm (fi A) = fi (Resm A),

for all sufﬁciently large m > |i|, where Resm 0 = 0. We set wt(A) = n0 + mi i ∈ P, where mi = (aij − δ−i,|i| ), i∈B

εi (A) = max{ k : eik A = 0 },

j∈Bn

ϕi (A) = max{ k : fik A = 0 }.

is a gl∞ -crystal, and the map w → A(w) is an isomorphism of gl∞ Then M∞×n since each column in A ∈ M∞×n can be identiﬁed with crystals from F⊗n to M∞×n a word in F. Next, let us deﬁne a gln -crystal structure on M∞×n . Since M∞×n can be identiﬁed ⊗n ⊗n and j ∈ In , with F , this will induce a gln -crystal structure on F . For A ∈ M∞×n ∗ ∗ deﬁne ej A and fj A to be the unique elements in M∞×n ∪ {0} satisfying

Resm (ej∗ A) = ej∗ (Resm A),

Resm (fj∗ A) = fj∗ (Resm A),

Crystal Graphs and the Combinatorics of Young Tableaux

for all sufﬁciently large m > 0. Set wt ∗ (A) = mj j ∈ Pn ,

where mj =

j∈Bn

ε∗j (A)

= max{ k :

501

(aij − δ−i,|i| ), i∈B

(ej∗ )k A

= 0 },

ϕj∗ (A)

= max{ k : (fj∗ )k A = 0 }.

Then M∞×n is a gln -crystal. Since the crystal structures on M∞×n are induced from those on M2m+1×n , it is clear that M∞×n is a (gl∞ , gln )-bicrystal. Suppose that A = (aij ) ∈ M∞×n is given and A = A(w) for w = w(1) ⊗ · · · ⊗ (n) ⊗n w ∈ F . Identifying a semi-inﬁnite word in F with a semi-inﬁnite semistandard tableau of a single column, we can deﬁne an algorithm of inserting a semi-inﬁnite word in F into another one in a similar way as in 4.3 so that we obtain a semi-inﬁnite semistandard tableau with two columns. Now we deﬁne the P-tableau of A to be

P(A) = (((· · · ( w(1) ← w(2) ) · · · ) ← w(n) ), which is a semi-inﬁnite semistandard tableau with n columns5 . Then P(A) is gl∞ crystal equivalent to A. Example 5.17. Let A be the matrix given in Example 5.16. Then

P(A) =

.. .. .. . . . −4 −4 −4 −3 −3 −1 −2 ← −1 ← 0 2 3 1 3 2 3

=

−4 −3 −2 −1 0 1 2 3

.. . −4 −4 −3 2 −1 3 . 3

Next, deﬁne the Q-tableau of A as follows. Choose a sufﬁciently large m such that aij = 1 for all i ≤ −m and aij = 0 for all i ≥ m, and let Qm be the unique semistandard tableau which is gln -crystal equivalent to Resm A, that is, Qm = Q(Resm A). Then the connected components of A and Qm are isomorphic as In -colored oriented graphs, but wt ∗ (A) = wt(Qm )−(m+1)(1 +· · ·+n ). So, if we deﬁne the Q-tableau of A to be Q(A) = σ −m−1 (Qm ) (see 3.4), then Q(A) is a rational semistandard tableau, which is gln -crystal equivalent to A. Note that Q(A) is independent of the choice of m. 5 More precisely, P(A) is the unique semi-inﬁnite semistandard tableau such that Res (P(A)) = m P(Resm (A)) for all m " 0, where P(A) is identiﬁed with its matrix form in M∞×n .

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Example 5.18. Let A be as in Example 5.16. Then ⎛ ⎞ 1 1 0 ⎜ ⎟ ⎜ 1 0 0 ⎟ ⎜ ⎟ ⎜ 0 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ Res3 A = ⎜ 0 0 1 ⎟ ∈ M7×3 , ⎜ ⎟ ⎜ 0 0 1 ⎟ ⎜ ⎟ ⎜ 1 0 1 ⎟ ⎝ ⎠ 1 1 1 and 1 Q3 = Q(Res3 A) = 2 3

1 2 3

1 2

1

3

3

3 .

Therefore, 3 Q(A) = σ −4 (Q3 ) =

−3 −3 −2

3

3 .

as a (gl∞ , gln )It can now be concluded that each connected component in M∞×n bicrystal is isomorphic to B(λ) × Bn (μ) for some λ and μ ∈ Zn+ . Moreover, A = is of highest weight if and only if (aij ) ∈ M∞×n

aij = 0 =⇒ ai+1 j = 0, and ai j+1 = 0 (j < n) (see 5.2). For a highest weight element A ∈ M∞×n , if λj is set to be λj = (aij − δ−i,|i| ) (j ∈ Bn ), i∈B

then λ = (λ1 , . . . , λn ) ∈ Zn+ and (P(A), Q(A)) = (H λ , Hnλ ). Summarizing, we obtain the decomposition of F⊗n and hence F ⊗n , which can be viewed as a semi-inﬁnite analogue of the Robinson–Schensted–Knuth correspondence. Theorem 5.19. The map w → (P(A(w)), Q(A(w))) gives the following isomorphism of (gl∞ , gln )-bicrystals; F⊗n B(λ) × Bn (λ). λ∈Zn+

Remark 5.20. The decomposition of the corresponding representation of Lie algebras was ﬁrst obtained by Frenkel [44] (see also [18]).

Crystal Graphs and the Combinatorics of Young Tableaux

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References [1] T.H. Baker, An insertion scheme for Cn crystals, Prog. Math. 191 (2000), 1–48. [2] G. Benkart, S.-J. Kang, M. Kashiwara, Crystal bases for the quantum superalgebra Uq (gl(m, n)), J. Am. Math. Soc. 13 (2000), 295–331. [3] V. Chari, A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994. [4] V.I. Danilov, G.A. Koshevoy, Bi-crystals and crystal (GL(V), GL(W)) duality, RIMS Preprint no. 1485, (2004). [5] V.I. Danilov, G.A. Koshevoy, Arrays and the combinatorics of Young tableaux, Russian Math. Surv. 60 (2005), 269–334. [6] E. Date, M. Jimbo, T. Miwa, Representations of Uq (gl(n, C)) at q = 0 and the Robinson-Schensted correspondence, in: L. Brink, D. Friedan, A.M. Polyakov eds., Physics and Mathematics of Strings, Memorial Volume of V. Knizhnik, World Scientiﬁc, 1990, 185–211. [7] V.G. Drinfel’d, Hopf algebras and the quantum Yang-Baxter equations, Soviet Math. Dokl. 32 (1985), 254–258. (1) [8] O. Foda, B. Leclerc, M. Okado, J.-Y. Thibon, T.A. Welsh, Branching functions of An−1 and JantzenSeitz problem for Ariki-Koike algebras, Adv. Math. 141 (1999), 322–365. [9] I. Frenkel, Representations of afﬁne Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, Lect. Notes Math. 933 (1982), 71–110. [10] W. Fulton, Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997. [11] T. Hayashi, q-analogues of Clifford and Weyl algebras—Spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129–144. [12] J. Hong, S.-J. Kang, Quantum groups and crystal bases, Graduate Stud. Math. 42, Amer. Math. Soc., 2002. [13] R. Howe, Remarks on classical invariant theory, Trans. Am. Math. Soc. 313 (1989), 539–570. [14] J.C. Janzten, Lectures on quantum groups, Graduate Stud. Math. 6, Amer. Math. Soc., 1996. [15] M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equations, Lett. Math. Phys. 10 (1985), 63–69. sl(n)) at q = 0, [16] M. Jimbo, K. Misra, T. Miwa, M. Okado, Combinatorics of representations of Uq ( Comm. Math. Phys. 136 (1991), 543–566. [17] V.G. Kac, Inﬁnite Dimensional Lie Algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. [18] V.G. Kac, A. Radul, Representation theory of the vertex algebra W1+∞ , Transform. Groups 1 (1996), 41–70. [19] S.-J. Kang, J.-H. Kwon, Tensor product of crystal bases for Uq (gl(m, n))-modules, Comm. Math. Phys. 224 (2001), 705–732. [20] S.J. Kang, M. Kashiwara, K. Misra, T. Miwa, T. Nakashima, A. Nakayashiki, Perfect crystal of quantum afﬁne Lie algebra, Duke Math. J. 68 (1992), 499–607. [21] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), 249–260. [22] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. [23] M. Kashiwara, Crystal bases and Littelmann’s reﬁned Demazure character formula, Duke Math. J. 71 (1993), 839–858. [24] M. Kashiwara, On crystal bases, Representations of groups, CMS Conference Proceedings,American Mathematics Society, Providence, RI, 1995, pp. 155–197. [25] M. Kashiwara, Realizations of crystals, Contemp. Math. 325 (2003), 133–139. [26] M. Kashiwara, T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 (1994), 295–345. [27] D. Knuth, Permutations, matrices, and the generalized Young tableaux, Paciﬁc J. Math. 34 (1970), 709–727.

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[28] J.-H. Kwon, Crystal graphs for Lie superalgebras and Cauchy decompositions, J. Algebraic Combin. 25 (2007), 57–100. [29] A. Lascoux, Double crystal graphs, Prog. Math. 210 (2003), 95–114. [30] A. Lascoux, M.P. Schützenberger, Le monoïde plaxique in non-commutative structures in algebra and geometric combinatorics, Quad. “Ricerca Sci.” 109 (1981), 129–156. [31] B. Leclerc, J.-Y. Thibon, The Robinson-Schensted correspondence, crystal bases, and the quantum straightening at q = 0, Electron. J. Combin. 3 (1996), no. 2, Research Paper 11, approx. 24 pp. (electronic). [32] C. Lecouvey, Schensted-type correspondence, plactic monoid, and jeu de taquin for type Cn , J. Algebra 247 (2002), 295–331. [33] C. Lecouvey, Schensted-type correspondences and plactic monoids for types Bn and Dn , J. Algebraic Combin. 18 (2003), 99–133. [34] P. Littelmann, Paths and root operators in representation theory, Ann. Math. 142 (1995), 499–525. [35] P. Littelmann, A plactic algebra for semisimple Lie algebras, Adv. Math. 124 (1996), 312–331. [36] G. Lusztig, Introduction to Quantum Groups, Progress in Math. 110, Birkhäuser, 1993. [37] I.G. Macdonald, Symmetric functuins and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995. [38] K. Misra, T. Miwa, Crystal base for the basic representation of Uq (sl(n)), Comm. Math. Phys. 134 (1990), 79–88. [39] H. Nakajima, t-analogs and q-characters of quantum afﬁne algebras of type An and Dn , Contemp. Math. 325 (2003), 141–160. [40] T. Nakashima, Crystal base and a generalization of Littlewood-Richardson rule for the clasical Lie algebras, Comm. Math. Phys. 154 (1993), 215–243. [41] B. Sagan, The Symmetric Group–Represenations, Combinatorial Algorithms, and Symmetric Functions, Wadworth and Brooks, Cole, 1991. [42] J.R. Stembridge, Rational tableaux and the tensor algebra of gln , J. Combin. Theory Ser. A 46 (1987), 79–120. [43] S. Sundaram, Tableaux in the representation theory, in: D. Stanton, Ed., Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 19, Springer-Verlag, New York, 1990, 191–225. [44] G.P. Thomas, On Schensted’s construction and the multiplication of Schur functions, Adv. Math. 30 (1978), 8–32.

Quivers and Representations∗ Xueqing Chen and Ki–Bong Nam Department of Mathematical and Computer Sciences, University of Wisconsin–Whitewater, Whitewater, WI53190, USA E-mail: [email protected]; [email protected]

Tomáˇs Pospíchal Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4, Canada E-mail: [email protected]

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Kac–Moody algebras and quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Quivers and their representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Quivers and their representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Representations of Q and problems in linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Representation types of quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. The path algebra of a quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Auslander–Reiten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Quadratic forms of quivers and algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Dimension vectors and positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Gabriel theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. BGP reﬂection functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Ringel–Hall algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Ringel–Hall algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Quantum groups and Ringel–Hall algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. PBW basis and canonical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Root vectors in quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. PBW basis and canonical basis for C () of ﬁnite type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Root vectors in C () of tame type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Kronecker quiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. PBW basis and canonical basis for C () of tame type . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Dedicated to Professor Hazewinkel on the occasion of his 65th birthday.

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00209-X 507

509 514 519 519 522 524 525 528 530 533 533 535 536 537 538 539 540 540 541 545 549

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7. Root categories, Kac–Moody algebras and elliptic Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. A construction of the Kac–Moody algebra g (C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. A construction of the elliptic Lie algebra g(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Guide to the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

550 550 555 559 559 559

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1. Introduction Why quivers? In this introductory section, we provide some motivation for the use of the language of quivers in mathematics. Quivers arise in many areas of mathematics, including representation theory, algebraic geometry, and differential geometry, Kac–Moody algebras, and quantum groups. A quiver is just a directed graph, and a representation of a quiver associates a vector space to each vertex and a linear map to each arrow. The notions of quiver and its representations were introduced by P. Gabriel in the early 1970s of the past century. The introduction of quivers marked the starting point of the modern representation theory of ﬁnite-dimensional associative algebras. A number of remarkable connections to other mathematical ﬁelds have been discovered, in particular to Hall algebras, quantum groups, elliptic Lie algebras, and more recently cluster algebras. In the following short overview of these topics, we highlight some of the main contributors. Representations of ﬁnite-dimensional algebras: (P. Gabriel, M. Auslander, I. Reiten, V. Dlab, C. M. Ringel, V. Kac et al.). The representation theory of ﬁnite-dimensional algebras can be formulated using the language of quivers. A basic ﬁnite-dimensional hereditary algebra over an algebraically closed ﬁeld k is isomorphic to a path algebra kQ associated to a quiver Q. If it is basic but not hereditary, it is isomorphic to a certain quotient algebra of a path algebra kQ, where the quiver Q depends only on homological properties of the simple modules of the algebra. In effect, the study of representations of a ﬁnite-dimensional algebra over an algebraically closed ﬁeld is reduced to the study of a quotient of a path algebra of a certain quiver. If the base ﬁeld is not algebraically closed, one may use “species” (or valued graphs) introduced by V. Dlab and C. M. Ringel in [1] instead of quivers to realize ﬁnite-dimensional algebras. Methods based on quivers and their representations have been used in the past 30 years extensively to describe the structure of length categories (Abelian categories where every object has a ﬁnite composition series), which arise very frequently in algebra. These techniques provide much better understanding of the indecomposable objects and sometimes allow a fairly complete description of the category. Ringel–Hall algebras: (C. M. Ringel, J. A. Green). Hall algebras were introduced by E. Steinitz and P. Hall in the context of counting subgroups of ﬁnite Abelian groups. In a broad sense, a Hall algebra is a tool allowing one to code a category. C. M. Ringel realized that Hall algebras provide a link between the representation theory of quivers on one side and Lie algebras and quantum groups on the other. This discovery had a profound inﬂuence on representation theory.

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The Ringel–Hall algebra is an algebra, which has a basis labeled by isomorphism classes of modules of a ﬁnite-dimensional hereditary algebra (or, more specially, representations of a quiver). The “structure constants” describing the multiplicative structure (relative to this basis) reﬂect the structure of the underlying modules. The Ringel–Hall algebra encodes a large amount of information about the modules of ﬁnite-dimensional hereditary algebra. Amazingly, the generic composition algebra of the Ringel–Hall algebra is isomorphic to the positive part of a quantum group. Via the Ringel–Hall algebra approach, J. Xiao and L. Peng [2] provided a realization of all symmetrizable Kac–Moody algebras, and Y. Lin and L. Peng [3] obtained a realization of some elliptic Lie algebras. Canonical bases of quantum groups: (G. Lusztig). A canonical basis of the quantum group Uv+ (g) was ﬁrst introduced by G. Lusztig in [4] via an elementary algebraic deﬁnition when g is a simple ﬁnite-dimensional Lie algebra. Later, he examined this basis in the framework of Ringel–Hall algebras. This canonical basis provides a natural setting for classical combinatorial results in the representation theory of simple Lie algebras. The original algebraic deﬁnition does not work for quantum groups of arbitrary Kac–Moody algebras. The difﬁculty in constructing the basis arises from the need to deﬁne suitable analogs of imaginary root vectors. By using topological methods [5], G. Lusztig constructed this canonical basis of the quantum group of a symmetric Kac–Moody algebra. It was shown that the “global crystal basis” introduced by M. Kashiwara and this canonical basis coincide. This canonical basis has very nice properties, which make it a powerful tool for the study of representation theory of quantum groups, Hecke algebras, quantized Schur algebra, etc. It is particularly suited for the investigation of the structure of integrable modules of quantum groups and their tensor products. Quiver variety: (G. Lusztig and H. Nakajima). By introducing quiver variety, G. Lusztig provided a geometric realization of the positive part of the universal enveloping algebra of a simply laced Kac– Moody algebra g using constructible functions. A geometric realization of the positive part of a quantum group using perverse sheaf permitted the construction of a canonical basis, which plays a major role in representation theory. The quiver varieties introduced by H. Nakajima yield a geometric realization of the highest weight representations of simply laced Kac–Moody algebras. Category O and stratiﬁed algebras: (E. Cline, B. Parshall, and L. Scott). Quasi-hereditary algebras were introduced by E. Cline, B. Parshall, and L. Scott in their efforts to understand Kazhdan–Lusztig theory and developed

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further by V. Dlab and C. M. Ringel. Quasi-hereditary algebras and their generalizations may be viewed as generalizations of quiver algebras more appropriate for understanding the relationship between standard modules and irreducible modules in the category O, perverse sheaves, and categories of representations of algebraic groups in positive characteristic. Some of the most interesting generalizations of quasi-hereditary algebras, called stratiﬁed algebras, have been deﬁned in independent (and slightly different) ways by I. Ágoston, V. Dlab and E. Lukács, and E. Cline, B. Parshall, and L. Scott. In each case, the stratiﬁcation involved occurs categorically at the derived category level and internally in terms of particular ideals of the algebra. Weighted projective lines and canonical algebras: (W. Geigle and H. Lenzing). W. Geigle and H. Lenzing in [6] introduced a new class of curves called weighted projective lines. These curves are essentially lines in a weighted projective space and behave like smooth projective curves with respect to coherent sheaves and vector bundles. In particular, the category of coherent sheaf coh C over a weighted projective line C has Serre duality, almost-split sequences, and each coherent sheaf splits into a direct sum of a vector bundle and a torsion sheaf. There is a bijective correspondence between weighted projective lines and canonical algebras as studied by C. M. Ringel in [7]. By means of a tilting sheaf, one has an equivalence Db (coh C) Db (mod −) of the derived categories of coh C and the category mod − of ﬁnite-dimensional –modules, where is the canonical algebra associated to C. In the case where the virtual genus of C is one, a brief account of the classiﬁcation of indecomposable bundles on C can be given. The equivalence Db (coh C) Db (mod −) establishes a link between M. F. Atiyah’s classiﬁcation of vector bundles on elliptic curves and C. M. Ringel’s classiﬁcation of modules over canonical algebras of tubular type. Dynkin diagrams of type A, D, and E: An

Dn E6

……

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E7

E8 The Dynkin diagrams An , Dn , E6 , E7 , and E8 have appeared in many different parts of mathematics. The A–D–E classiﬁcations: In accordance with the ubiquity of A–D–E classiﬁcations [8, 9], the following graph shows some of the main occurrences of Dynkin diagrams in mathematics.

Representation Theory of f.d. Algebras

Kac–Moody Lie Algebras

Quiver Representations Gabriel Theorem

3-dimensional Orthogonal Groups

Singularities of Algebraic Hypersurfaces

Quantum Groups

Semisimple Lie Algebras and Lie Groups

Dynkin Diagrams

Simple Groups of Lie Type Chevalley Groups

Critical Points of Smooth Functions

As V.I. Arnold said in [8]: The problem is to ﬁnd a common origin of all A–D–E classiﬁcation theorems and to substitute a priori proofs to a posteriori veriﬁcations of the parallelism of the classiﬁcations. Are there direct relationship among some ﬁelds of mathematics where the same diagrams occur? Can results already developed in one ﬁeld be applied to get new result and new information in another ﬁeld? In the following sections, you will see why these diagrams have come up in the representation theory of ﬁnite-dimensional algebras and related topics and how the representation theory of ﬁnite-dimensional algebras is connected with other ﬁelds of mathematics.

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Classes of Lie algebras: The class of semisimple Lie algebras has been extended to several more general classes. We list some of those related to the topic of this chapter without mentioning details.

Extended affine Lie algebras

Toroidal Lie algebras

Affine Kac–Moody Lie algebras Finite-dimensional Semi-simple Lie algebras

Symmetrizable

Kac–Moody

Kac–Moody

Lie algebras

Lie algebras

The ﬁnite-dimensional algebras can be classiﬁed with respect to their representation types. Every algebra is either of ﬁnite type, tame type, or wild type. In this chapter, the close connection between the classiﬁcation of ﬁnite-dimensional algebras and the classiﬁcation of Kac–Moody algebras will be given. The main problems of the representation theory of ﬁnite-dimensional algebras are: • to study the structure of the given algebra; • to describe all indecomposable modules and their homological properties; • to classify the given algebra. The following graph (c.f. [10]) illustrates the relations between the representation theory of quivers and Lie theory, and also provides a guideline for the following sections. g

g

n

h

n

Dynkin

g

n

h

n

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In Section 2, we recall the basic terminology of Kac–Moody algebras and quantum groups, which is needed in the following sections. We introduce the language of quivers and their representations in Section 3 and describe the relation between quivers and ﬁnite-dimensional algebras. Section 4 provides the ﬁrst connection between the representation theory of quivers and Lie theory; Gabriel theorem is explained. We give the deﬁnition of the Ringel–Hall algebra and discuss the amazing isomorphism between the generic composition algebra and the positive part of the corresponding quantum group in Section 5. In Section 6, by using the Ringel–Hall algebra approach, we construct root vectors, the Poincaré–Birkhoff–Witt (PBW) basis, and the canonical basis of a quantum group of ﬁnite or afﬁne type. Section 7 provides a realization of all symmetrizable Kac–Moody algebras and some elliptic Lie algebras (1,1) (1,1) (1,1) (1,1) via the root category of hereditary algebras of type D4 , E6 , E7 , and E8 and tubular algebras of the same type, respectively. f

g

Throughout this chapter, the composition of any two maps X → Y → Z is denoted by gf and the set of non-negative integers is denoted by N.

2. Kac–Moody algebras and quantum groups We will use the Cartan datum to unify a variety of set-ups in different mathematical ﬁelds. Deﬁnition 2.0.1 (G. Lusztig). A Cartan datum is a pair = (I, ( , )) where • I is a ﬁnite set; • ( , ) is a symmetric bilinear form on ZI . It is assumed that • (i, i) ∈ {2, 4, 6, . . . } for any i ∈ I; • 2

(i, j) ∈ {0, −1, −2, . . . } for any i = j in I. (i, i)

A Cartan datum is symmetric Cartan datum if (i, i) = 2 for all i ∈ I; we call simply laced if it is symmetric and (i, j) ∈ {0, −1} for all i = j. A Cartan datum is said to be irreducible Cartan datum if I is nonempty and for any i = j in I, there exists a sequence i = i1 , i2 , . . . , im = j in I such that (ip , ip+1 ) < 0 for p = 1, 2, . . . , m − 1. A matrix C is called symmetrizable matrix if there exists an invertible diagonal matrix D such that DC is symmetric. Note that any symmetric matrix is symmetrizable. Every Cartan datum determines a symmetrizable (generalized) Cartan matrix in (i, j) the following way: write aij = 2 , and let C = (aij )i,j∈I . In the following until (i, i) the end of this section, assume I = {1, . . . , n}.

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Deﬁnition 2.0.2. A realization of a symmetrizable (generalized) Cartan matrix C = (aij )1≤i,j≤n is a triple (h, , ∨ ), where h is a complex vector space, = {α1 , α2 , . . . , αn } ⊂ h∗ , and ∨ = {h1 , h2 , . . . , hn } ⊂ h such that (1) both and ∨ are linearly independent; (2) dim h = 2n − rank(A); (3) hi , αj = aij for all i, j ∈ I, where , : h × h∗ −→ C, h, α = α(h). Deﬁnition 2.0.3. Let C = (aij )1≤i,j≤n be a symmetrizable (generalized) Cartan matrix. The associated Kac–Moody algebra g = g(C) is deﬁned as the complex Lie algebra with generators ei , fi (i = 1, . . . , n) and h and relations: [h, h ] = 0,

∀ h, h ∈ h;

[ei , fj ] = δij hi ; [h, ei ] = αi (h)ei ,

∀ h ∈ h;

[h, fi ] = −αi (h)fi ,

∀ h ∈ h;

(ad ei

= 0,

if i = j;

(ad fi )1−aij fj = 0,

if i = j,

)1−aij e

j

where ad ei = [ei , −]. The elements ei and fj are called Chevalley generators. Let n+ , respectively n− , be the Lie subalgebra of g generated by {ei |1 ≤ i ≤ n}, respectively {fi |1 ≤ i ≤ n}. The Kac–Moody algebra g has triangular decomposition: g = n− ⊕ h ⊕ n+ . Denote by g = g (C) the subalgebra of g generated by all Chevalley generators ei and fj . Theorem 2.0.4 [11]. Let h ⊂ h be the span of h1 , . . . , hn . Then (1) g = n− ⊕ h ⊕ n+ ; (2) g = [g, g]. Remark 2.0.5. The algebra g = g (C) is the derived algebra of a Kac–Moody algebra g. The algebra g can be deﬁned as the complex Lie algebra with 3n generators ei , fi , hi (1 ≤ i ≤ n) and the following deﬁning relations: [hi , hj ] = 0,

∀ i, j;

[ei , fj ] = δij hi ; [hi , ej ] = aij ej ; [hi , fj ] = −aij fj ; (ad ei )1−aij ej = 0,

if i = j;

(ad fi )1−aij fj = 0,

if i = j.

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Note that the Kac–Moody algebra g and its derived algebra g share a common structure and representation theory. Any element h ∈ h gives rise to a decomposition of g into adh -eigenspaces, and since h is commutative, there is a simultaneous such decomposition g = ⊕α∈h∗ gα , where gα = {x ∈ g | [h, x] = α(h)x, ∀h ∈ h}. A nonzero element α ∈ h∗ such that gα = {0} is called a root of g, and the set of roots = {α ∈ h∗ | gα = {0}} is called the root system of g (relative to h). The number mult α := dim gα is called the multiplicity of α. Note that = {α1 , . . . , αn } ⊂ . Every root β ∈ can be written as β=

n

li αi

i=1

with integer coefﬁcients li , all non-negative or all nonpositive. The root in is then called simple root. If all li ≥ 0 (respectively, all li ≤ 0), we call β positive root (respectively negative root). The collections of positive and negative roots will usually be denoted by + and − . The set comes with a canonical polarization = + − and + = −− . For each i = 1, . . . , n, deﬁne the simple reﬂection si of h∗ by the formula si (ι) = ι − hi , ι αi ,

ι ∈ h∗ .

Note that si ∈ GL(h∗ ). The subgroup W = W(C) of GL(h∗ ) generated by all simple reﬂections is called the Weyl group of g. A root α belonging to W · is called real root and obviously satisﬁes dim gα = 1. A root that is not real is called imaginary root and may have higher multiplicity. The imaginary roots are W–invariant, which means W · img = img . Let + real be the set be the set of all positive imaginary roots. We know of all positive real roots and + img + + that every positive root is either real or imaginary, thus we have = real + img . For simplicity, we only consider case of an irreducible and simply laced Cartan datum = (I, ( , )) in the following sections, except Section 7. If is symmetric and irreducible, then the corresponding Cartan matrix is symmetric and indecomposable. We can associate a graph of a symmetric Cartan matrix. Its vertices are labeled by 1, 2, . . . , n (n = |I|). If i = j, the vertices i,j are jointed by aij aji edges. Recall that a Cartan matrix is called indecomposable if its associated graph is connected.

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Theorem 2.0.6. Let g(C) be a Kac–Moody algebra with C symmetric and indecomposable. Then, • g(C) is a simple Lie algebra ⇐⇒ the graph of C is one of Dynkin diagrams An , Dn , E6 , E7 , and E8 . • g(C) is afﬁne Kac–Moody algebra ⇐⇒ the graph of C is one of Euclidean n , diagrams An , D E6 , E7 , and E8 . The Dynkin diagrams of type A, D, and E have been displayed in the previous section. The Euclidean diagrams (called by some authors “extended Dynkin diagrams”) and of type A, D, E are as follows:

An

Dn

E6

E7

E8 Quantum group of type = (I, ( , )). Quantum groups were initially developed in the context of the inverse scattering methods and used to solve integrable quantum systems. V. Drinfel’d and M. Jimbo showed independently in 1985 that quantum groups are Hopf algebras that are often deformations of universal enveloping algebras of Lie algebras. Since then, quantum groups have brought a new level of understanding to a wide variety of areas of physical and mathematical research. In mathematics, they have been used to develop invariants of knots; other applications have been found in category theory and also in the representation theory of algebraic groups in prime characteristic. They are very inﬂuential in modern developments of ring theory and provide an inspiration to the new ﬁeld of quantum geometry.

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The Cartan datum considered in this chapter is nothing but a symmetrizable generalized Cartan matrix together with its W -invariant scalar product (, ) deﬁned on ZI , where W is the Weyl group and ZI is the lattice generated by the root system . Let g = g(C) be a symmetrizable Kac–Moody Lie Algebra of type = (I, (, )), where = (I, (, )) is a Cartan datum. Let Q(v) be the ﬁeld of rational function in the variable v. Deﬁnition 2.0.7 (Drinfel’d–Jimbo). The quantum group (or “quantized enveloping algebra”) U = Uv (g) of type is deﬁned as a Q(v)-algebra with generators Ei , Fi , Ki , and K−i , i ∈ I subject to the relations: Ki K−i = 1 = K−i Ki , Ki Kj = Kj Ki , Ki Ej = v(i,j) Ej Ki , Ki Fj = v−(i,j) Fj Ki , Ei Fj − Fj Ei = δij

Ki − K−i vi − v−1 i

for any i, j ∈ I, where vi = v 1−aij

(−1)

t

1 − aij t

t=0 1−aij

(−1)t

t=0

1 − aij t

(i,i) 2

εi

εi

, and 1−aij −t

=0

1−aij −t

=0

Eit Ej Ei Fit Fj Fi

for any i = j in I, where aij = 2 (i,j) (i,i) , εi = [s]εi = [s]εi ! = s r

εi

=

(i,i) 2 ;

for i ∈ I, we use the notation

(vεi )s − (vεi )−s = (vεi )s−1 + (vεi )s−3 + · · · + (vεi )−s+1 , (vεi ) − (vεi )−1 s r=1

[r]εi , and

[s]εi ! [r]εi ![s − r]εi !

here, r and s are non-negative integers, and r ≤ s. For convenience, we set [0]εi ! = 1. s s = . In the case when εi = 1, write [s] = [s]1 , [s]! = [s]1 !, and r r 1

Quivers and Representations

Let A = Z[v, v−1 ]. Deﬁne the divided powers subalgebra A-subalgebra of Uv (g) generated by the elements of the set (r)

{K±i , Ei =

519 AU

to be the

Eir Fr (r) , Fi = i |i ∈ I, r ≥ 0}. [r]εi ! [r]εi !

We deﬁne A U± in an analogous manner. Next, we deﬁne the bar involution ( ) of Uv (g) by Ei = Ei , Fi = Fi , v = v−1 . The involution preserves both U± and A U± .

3. Quivers and their representations We will work over a base ﬁeld k. Unless mentioned otherwise, k is assumed to be algebraically closed. 3.1. Quivers and their representations Deﬁnition 3.1.1. As mentioned before, a quiver is a directed graph. Formally, a ﬁnite quiver Q = (Q0 , Q1 , s, t : Q1 −→ Q0 ) is given by • Q0 = a ﬁnite set of vertices = {1, 2, . . . , n}; ρ • Q1 = a ﬁnite set of arrows = {ρ : s(ρ) −→ t(ρ)}, where the two maps s and t assign to each arrow ρ ∈ Q1 its source s(ρ) ∈ Q0 and its target t(ρ) ∈ Q0 , respectively. Note that multiple arrows between the same pair of vertices are allowed, as well as loops (arrows which start and end at the same vertex). The quiver Q is said to be connected quiver if the underlying graph (which is obtained from Q by forgetting about the orientation of the arrows) is a connected graph. We only consider connected quiver in the following. A nontrivial path σ in a quiver Q is a sequence σ = σ1 σ2 . . . σm of arrows, which satisﬁes t(σi ) = s(σi+1 ) for all 1 ≤ i ≤ m − 1. Often, we shall denote it by a diagram like the following: σ1

σ2

σm

σ = • → • → • · · · → •. This path starts at s(σ1 ) and terminates at t(σm ), m is the length of a path σ. We denote by s(σ) and t(σ) the starting and terminating vertex of the path σ. For each vertex i,

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we denote by ei the trivial path of length zero, which starts and terminates at i. A nontrivial path σ is called an oriented cycle if s(σ) = t(σ). 1 - 2 β- 3 Q: • α • • has three vertices and two arrows.

Example 3.1.2. The quiver

Remark 3.1.3. It is trivial to see that every simply laced Dynkin diagram can be made into a quiver by assigning an orientation to each edge. Note that there are lots of ways of orienting the edges of a Dynkin diagram to make a quiver. For example, the following two quivers 1

2

3

1

2

3

are both obtained from the same Dynkin diagram of type A3 . Given a quiver Q, we now deﬁne Rep(Q), the category of representations of a quiver Q. We will restrict our discussion to ﬁnite-dimensional representations of a quiver. The ingredients of Rep(Q) are given as follows: • objects: a (ﬁnite dimensional) representation V of a quiver Q is given by a ﬁnitedimensional k-vector space Vi for each i ∈ Q0 and a linear map Vρ : Vs(ρ) → Vt(ρ) for each ρ ∈ Q1 ; • morphisms: θ : V → V is given by linear maps θi : Vi → Vi for each i ∈ Q0 satisfying

θ

φ

• composition: V −→ V −→ V is given by (φθ)i = φi θi ; • the direct sum V ⊕ V of two representations of a quiver Q is given by (V ⊕ V )i = Vi ⊕ Vi , (V ⊕ V )ρ : Vs(ρ) ⊕ Vs(ρ)

i ∈ Q0 ,

Vρ 0 0 Vρ −−−−−−−−−→ Vt(ρ) ⊕ Vt(ρ) , ρ ∈ Q1 ;

• V is a trivial (or zero) representation if Vi = 0 for each i ∈ Q0 and Vρ = 0 for each ρ ∈ Q1 ; • V is indecomposable representation of a quiver Q if V V ⊕ V , where V and V are nontrivials;

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• the dimension vector of a representation of a quiver Q is denoted by dim V = (dimk Vi )i∈Q0 ; • for each i ∈ Q0 , the simple representation Si of a quiver Q is given by k when j = i Sij = and Siρ = 0 for all ρ ∈ Q1 . 0 otherwise It is easy to see that the simple representation has a standard vector as dimension vector, namely i

αi = dim Si = (0, 0, . . . , 1, . . . , 0, 0). 1

Example 3.1.4. Consider the quiver Q : • [0]

[1]

- •2

- •3

A representation V1 is given by 0 → k → k and another representation V2 is [1]

[1]

given by k → k → k. Both are ﬁnite dimensional and indecomposable. A morphism V1 → V2 is given by

[0,1]

[1]

It is easy to compute that the following representation k2 −−→ k −→ k is decomposable and [0,1]

[1]

k2 −−→ k −→ k

[1]

[1]

k −→ k −→ k

⊕

[0]

[0]

k −→ 0 −→ 0.

Moreover, we can see that [0,1]

[1] dim k2 −−→ k −→ k = (2, 1, 1) = 2α1 + α2 + α3 . Theorem 3.1.5 (Krull–Remak–Schmidt Theorem). Every ﬁnite-dimensional representation V of a quiver Q is isomorphic to a direct sum of indecomposable representations. This decomposition is unique up to isomorphism and permutation of factors. Thus, the problem of classifying representations of a quiver requires us to classify all indecomposable representations and then to ﬁnd what morphisms between those indecomposables can exist. We summarize the main problems of the representation theory of quivers as follows (cf. [12]): • to develop methods for constructing indecomposable representations; • to look for suitable invariants to identify indecomposable representations;

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• to show that a given list of indecomposable representations is complete; • to ﬁnd connections with other areas of mathematics. 3.2. Representations of Q and problems in linear algebra A whole range of problems of linear algebra can be formulated in a uniform way in terms of representations of quivers. 2 1 Example 3.2.1 (Equivalence of matrices). Consider the quiver Q : • α- • In this case, a ﬁnite-dimensional representation is given by a linear map f : V2 → V1 , the map itself may be given by a matrix A. The following diagram illustrates the situation when this representation and another one (given by a matrix B) are equivalent (i.e. isomorphic in the category of representations). The isomorphism is realized by two invertible matrices P and Q:

km

A

P km

kn Q

B

A

kn

The classiﬁcation of equivalence classes of linear maps amounts to classifying the n × m matrices up to “row and column equivalence.” As is well known from elementary linear algebra, the equivalence classes of matrices are completely determined by their size and rank. Every representation of dimension vector (m, n) is a direct sum [0]

[0]

of three types of indecomposable representations: S1 : 0 −→ k, S2 : k −→ 0, and [1]

S2,1 : k −→ k. The n × m matrix has rank r when the summand S2,1 occurs r times in its decomposition. The classiﬁcation of representations of this quiver is trivial since we only have three indecomposable representations up to isomorphism. 1 Example 3.2.2 (Similarity of matrices). Consider the quiver Q : •? α Giving a ﬁnite-dimensional representation simply means giving an endomorphism of a ﬁnite-dimensional vector space f : V1 → V1 . The following diagram illustrates the situation where two representations are isomorphic: kn

A

P kn

kn P

B

A

.

kn

The classiﬁcation of isomorphism classes is equivalent to the classiﬁcation of matrices up to conjugation. This problem is answered by the well-known Jordan classiﬁcation. In terms of indecomposable representations, this means that

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523

for a dimension vector (n), there is a one parameter family of indecomposable endomorphisms ⎛ ⎞ λ 1 0 ... 0 0 ⎜ 0 λ 1 ... 0 0⎟ ⎜ ⎟ . . . . . . . . . . . . . . . . . .⎟ Jn (λ) = ⎜ ⎜ ⎟ , λ ∈ k. ⎝ 0 0 0 ... λ 1⎠ 0 0 0 ... 0 λ n×n Example 3.2.3 (Equivalence of matrix pairs). Consider the Kronecker quiver β2 Q: - •1 • α A representation means a pair of linear maps: fα : V2 → V1 and fβ : V2 → V1 . Two representations are isomorphic if the following holds: km P km

A1 A2 B1 B2

kn Q kn

The problem of classifying such pairs of matrices was considered by K. Weierstrass, but his solution was incomplete. The problem was completely solved by L. Kronecker in 1890. Here, we list isomorphism classes of all indecomposable representations:

This time there are inﬁnitely many indecomposables, but we can classify them. An effective classiﬁcation of the representations of this quiver is possible. Example 3.2.4 (The ﬁve subspace problem). Consider the quiver:

A representation of this star quiver is given by V = {V1 , V2 , V3 , V4 , V5 , V6 ; fαi : Vi → V6 , 1 ≤ i ≤ 5}.

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If all maps are injective, we can have such a representation as a subspace conﬁguration. There exists a two-parameter family of pairwise nonisomorphic indecomposable representations, Vλ,μ = k, k, k, k, k, k2 ; 10 , 01 , 11 , λ1 , μ1 . Unfortunately, one cannot effectively ﬁnd a complete list of all indecomposable representations. The complete classiﬁcation of representations of this quiver seems impossible. 3.3. Representation types of quivers Based on the above discussion, we can distinguish quivers with respect to their representation types. The representation type of a quiver Q is one of the following three: ﬁnite type: the number of isomorphism classes of indecomposable representations is ﬁnite; tame type: the number of isomorphism classes of indecomposable representations is inﬁnite, but a complete list of indecomposables can be described explicitly. In each dimension, there exist only ﬁnitely many one-parameter families of pairwise nonisomorphic indecomposables; wild type: the quiver is neither of tame nor of ﬁnite type. As it turns out, in this case one cannot provide a “complete list” of indecomposable representations. We do not provide the precise deﬁnitions of tame and wild, as those are rather technical. It is well-known that every quiver is either of ﬁnite, tame, or wild type. Remark 3.3.1. The representation types of the quivers discussed in the examples are as follows: α- •1

2

Example 3.2.1 is of ﬁnite type:

•

Example 3.2.3 is of tame type:

2

1

1

Example 3.2.4 is of wild type: 5

2

6

4

3

The basic result in representation theory of quivers is the following theorem classifying the quivers with respect to different representation types.

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525

Theorem 3.3.2 [1, 13]. Let Q be a quiver with no oriented cycles, and let be the underlining graph of Q. We get this graph when we forget the orientation of arrows in Q. Then, • Q is of ﬁnite type ⇐⇒ is Dynkin. • Q is of tame type ⇐⇒ is Euclidean. • All other quivers are of wild type. 3.4. The path algebra of a quiver Starting with a quiver, one can construct an algebra that has some remarkable properties. Deﬁnition 3.4.1. Let Q be a quiver. The path algebra kQ is the k-vector space with • basis: {all paths in Q}; • multiplication given by: obvious composition στ = 0

if t(σ) = s(τ), otherwise.

This product is extended to the entire space kQ using distributivity and linearity. In this way, one obtains an associative algebra kQ. The “zero-length” path starting and ending at the vertex i is denoted by ei . Note that products are read from left to right (unlike composition of morphisms). 1 2 3 Example 3.4.2. Consider the quiver Q : • α - • β - • The path algebra kQ has basis {e1 , e2 , e3 , α, β, αβ}. A couple of examples of products are: βα = ββ = αβe1 = e3 αβ = 0; αβe3 = αβ. ? α , then Example 3.4.3. If Q consists of one vertex and one loop, that is, Q : • kQ k[T ]. If Q has one vertex and n loops, then kQ is the free associative algebra on n letters.

The basic properties of the path algebra kQ are dictated by the quiver Q. Proposition 3.4.4. Set = kQ. Note the following facts (cf. [14]) : (1) the set {ei | i ∈ Q0 } is a complete set of primitive orthogonal idempotents of ; n ei ; (2) has an identity given by i=1

(3) the spaces ei , ej , and ei ej have as bases the paths starting at i, terminating at j and starting at i and terminating at j, respectively; (4) distinct ei ’s are not equivalent, that is, ei ej for i = j;

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(5) = ⊕ni=1 ei , and each ei is an indecomposable projective right -module; (6) is ﬁnite dimensional ⇐⇒ Q has no oriented cycles; (7) if Q has no oriented cycles, then is a hereditary algebra. Recall that an algebra (or more generally, a ring) is called hereditary if every submodule of a projective module is projective. The following theorem tells us that we can identify ﬁnite-dimensional representations of a quiver Q with ﬁnitely generated (ﬁnite dimensional) right kQ-modules. Theorem 3.4.5. We have the following equivalence: Rep(Q) mod −kQ. Proof. We only provide the construction. If M is a kQ-module, deﬁne a representation V with Vi = Mei and Vρ (x) = xρ = xρet(ρ) ∈ Vt(ρ) . If V is a representation, deﬁne a module M as follows. Let M = i∈Q0 Vi . Let εi

πi

Vi −→ M −→ Vi be the canonical maps. Then, xei = εi πi (x), xσ1 σ2 . . . σm = εt(σm ) Vσm . . . Vσ2 Vσ1 πs(σ1 ) (x).

A module M over kQ can be identiﬁed with a representation of Q, which is given by a set of k-vector spaces indexed by Q0 along with a set of linear maps indexed by Q1 . Thus, M = (Mi , fα )i∈Q0 ,α∈Q1 , where fα ∈ Homk (Mi , Mj ) if α is an arrow from i to j. For instance, let Si be the simple representation corresponding to a vertex i, then it can be identiﬁed with the module, still denoted by Si = (Sij , fα ) where Sij = k, if j = i, Sij =0 , if j = i, fα = 0 for all α ∈ Q1 . Then, Si is a simple module and, in the case that kQ is ﬁnite dimensional, any simple module is of this form. Remark 3.4.6. Dealing with representations and morphisms of representations of a quiver Q amounts to the same thing as dealing with modules and homomorphisms of modules over the path algebra kQ. It is very easy to compute with representations since only vector spaces and matrices are involved. In the following, we will not distinguish between representations of a quiver and modules of corresponding path algebra. Given a ﬁnite-dimensional k-algebra A, consider it as a right module over itself and decompose it into indecomposables = ⊕ni=1 Pi , which are projective as is easily seen. The algebra A is called basic if Pi is not isomorphic to Pj for i = j. Note that every ﬁnite-dimensional algebra is Morita equivalent to a basic ﬁnite-dimensional algebra. The next two theorems elucidate the relationship of ﬁnite-dimensional associative algebras and path algebras of quivers. Theorem 3.4.7. A basic ﬁnite-dimensional hereditary algebra over an algebraically closed ﬁeld k is isomorphic to the path algebra kQ for some quiver Q.

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527

Since there are ﬁnite-dimensional k-algebras that are not hereditary, not every basic ﬁnite-dimensional k-algebra is isomorphic to the path algebra of a quiver. The following theorem is one of the cornerstones of the theory. Theorem 3.4.8. A basic ﬁnite-dimensional algebra over an algebraically closed ﬁeld k is isomorphic to a certain quotient algebra of the path algebra kQ for some quiver Q. Remark 3.4.9. A quotient algebra of a path algebra can be described by admissible relations. An admissible relation on Q over a ﬁeld k is a k-linear combination of paths of length ≥ 2. Without loss of generality, we can assume that all paths involved in one relation have the same starting vertex and the same terminating vertex. Given a set S of admissible relations, consider the kQ-ideal I generated by S, and call such an admissible ideal. The quotient algebra kQ/I can be ﬁnite dimensional even if kQ is inﬁnite dimensional. Just as before, a module over kQ/I can be identiﬁed with a representation of Q, where the set of maps is to satisfy the additional conditions that the linear combinations of homomorphisms corresponding to relations in S are zero. Let us reiterate that it is well known that any basic algebra over algebraically closed ﬁeld k is isomorphic to kQ/I for some quiver Q and some admissible ideal I. Denote by Rep(Q, I) the full subcategory of Rep(Q) consisting of the representations of Q bounded by I (see [7]). Then we have Rep(Q, I) mod −kQ/I. Example 3.4.10 (Quiver with zero relations). Consider the quiver

Choose the ideal I =< αβ − γδ > generated by one admissible relation. The quotient algebra kQ/I has a basis {e1 , e2 , e3 , e4 , α, β, γ, δ, αβ = γδ}. It is well known that any ﬁnite-dimensional algebra is Morita equivalent to precisely one basic ﬁnite-dimensional algebra. Thus, most problems from the representation theory of ﬁnite-dimensional algebras can be understood as problems about representations of quivers. Remark 3.4.11. In case k is not algebraically closed ﬁeld, one may have to resort to the use of species (also known as a “modulated quiver”) instead of a simple quiver. The interested readers may consult [1]. The representation type of an algebra is deﬁned in the same way as it was done for quivers. Alas, the rigorous deﬁnitions of tame and wild type make it far from obvious

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that each algebra is precisely one of the three types. Fortunately, this highly nontrivial fact is true. Theorem 3.4.12 [15]. Let A be a ﬁnite-dimensional k-algebra. Then A is of ﬁnite, tame, or wild representation type.

3.5. Auslander–Reiten theory Let A be a ﬁnite-dimensional k-algebra. We would like to record information on the category mod −A of right A–modules in terms of a directed graph. It seems most appropriate that vertices should represent some modules and arrows should represent some homomorphisms. Since mod −A is a Krull–Remak–Schmidt category, we only allow one point for each isomorphism class of indecomposable modules. As for the arrows, they should stand for “irreducible homomorphism,” that is, those with no “nontrivial” factorization. Deﬁnition 3.5.1. A homomorphism f : X → Y in mod −A is irreducible morphism provided (a) f is neither a split monomorphism nor a split epimorphism, and (b) if f = f1 f2 , either f1 is a split epimorphism or f2 is a split monomorphism, where f2 : X → Z and f1 : Z → Y . Deﬁnition 3.5.2. A homomorphism f : X → Y in mod −A is right almost split morphism provided (a) it is not a split epimorphism, and (b) any morphism Z → Y , which is not a split epimorphism factors through f . The deﬁnition of a left almost split morphism is dual. f

g

Deﬁnition 3.5.3. An exact sequence 0 −→ X −→ Y −→ Z −→ 0 in mod −A is called an almost split sequence (also called Auslander–Reiten sequence) if f is left almost split and g is right almost split. Almost split sequences can be characterized in terms of irreducible morphisms. f

g

Theorem 3.5.4 [16]. An exact sequence 0 −→ X −→ Y −→ Z −→ 0 is an almost split sequence if and only if f and g are both irreducible. We can construct almost split sequences in mod −A through a functor. First, consider the A-dual functor (−)t = HomA (−, A) : mod −A → mod −Aop ,

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529

where mod −Aop is the category of all ﬁnite-dimensional left A-modules. Start by approximating each right A-module M by projective modules. Let p1

p0

→0 P1 −→ P0 −→ M − be a minimal projective presentation of M, that is, an exact sequence such that p0 : P0 −→ M and p1 : P1 −→ Ker p0 are projective covers. Applying the left exact and contravariant functor (−)t , we obtain an exact sequence of left A-modules pt0

pt1

0 −→ M t −→ P0t −→ P1t −→ Coker pt1 −→ 0. Denote Coker pt1 by Tr(M) and call it the transpose of M. The transpose functor transforms right A-module into left A-module. Thus, if we wish to deﬁne an endofunctor of mod −A, we need to compose it with another duality between right and left A-modules, namely the standard duality D = Homk (−, k). The Auslander–Reiten translations are deﬁned to be the compositions of D and Tr, namely, set τ = DTr and τ −1 = TrD. The existence theorem of almost split sequences is described as follows. Theorem 3.5.5 [16]. Let A be a ﬁnite-dimensional k-algebra. (a) For any indecomposable nonprojective A-module M, there exists an almost split sequence 0 −→ τM −→ E −→ M −→ 0; (b) For any indecomposable noninjective A-module N, there exists an almost split sequence 0 −→ N −→ F −→ τ −1 N −→ 0. In the following, a device for studying all left and right almost split morphisms simultaneously is introduced. Deﬁnition 3.5.6. The Auslander–Reiten quiver (A) of mod −A is constructed as follows: • the set of vertices 0 consists of the isomorphism classes [M] of ﬁnitedimensional indecomposable modules; • let [M], [N] be the vertices in (A) corresponding to the indecomposable modules M, N in mod −A. The arrows [M] → [N] are in bijective correspondence with the vectors of a basis of the k-vector space Irr(M, N) (all irreducible morphisms from M to N). The Auslander–Reiten quiver of A encodes a lot of information about the homological properties of mod −A. Now, (A) can be decomposed into components in a straightforward way: two objects X and Y lie in the same component if and only if there exist a natural number n, indecomposable modules Xi , for 1 ≤ i ≤ n, and for each i, either an irreducible

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morphism fi : Xi → Xi+1 or an irreducible morphism gi : Xi+1 → Xi with X1 = X and Xn = Y . Theorem 3.5.7. Let Q be a quiver without oriented cycles. Then, kQ is of ﬁnite representation type if and only if (kQ) is a ﬁnite connected quiver (i.e. has only a ﬁnite connected component). - •2

1

Example 3.5.8. Consider the quiver Q : • Reiten quiver (kQ) looks as follows: k

0

0

0

k

k

k

k

k

k

0

k

- •3 . The Auslander–

0

k

0

k

0

0,

where the dotted lines indicate the Auslander–Reiten translation. The Auslander– Reiten quiver encodes much information about the representations and morphisms between them. Remark 3.5.9. Given a quiver Q without oriented cycles, let = kQ be the path algebra. If it is of ﬁnite type, then one can construct all the indecomposables systematically and completely by following the “ﬂow” of the Auslander–Reiten quiver (). Start with indecomposable projectives ei (or, alternatively, the injectives Homk (ei , k)), for all i ∈ Q0 , which correspond to the vertices of the quiver Q, and then apply the inverse of the Auslander–Reiten translation functor τ −1 (or τ) repeatedly to get all the indecomposables. 3.6. Quadratic forms of quivers and algebras Deﬁnition 3.6.1. Let A be a basic ﬁnite-dimensional k-algebra (associative, with unit) and let S1 , S2 , . . . , Sn be a complete set of isomorphism classes of simple right A-modules. Given any A-module M, the dimension vector of M is the vector dim M =

n [M : Si ]αi , i=1

where [M : Si ] is the Jordan–Hölder multiplicity of Si in M and αi is the standard coordinate vector (or, equivalently, the dimension vector of the simple module Si ).

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Remark 3.6.2. Let Q be a quiver without oriented cycles. Let M be an kQ-module. The dimension vector of M is the vector [M : Si ]αi . dim M = (dimk Mei )i∈Q0 = i∈Q0

Deﬁnition 3.6.3. Let A be a k-algebra. The Grothendieck group of A is the Abelian group G(A) = F/F , where F is the free Abelian group having as basis the set of the isomorphism classes [M] of modules M in mod −A and F is the subgroup of F generated by all elements [M] − [L] − [N] corresponding to an exact sequence 0 −→ L −→ M −→ N −→ 0. Theorem 3.6.4. Let A be a basic ﬁnite-dimensional k-algebra and let S1 , S2 , . . . , Sn be a complete set of isomorphism classes of simple right A-modules. Then there exists an isomorphism dim : G(A) −→ Zn such that dim([M]) = dim M. Note that G(A) can be identiﬁed with Zn . Proposition 3.6.5. Let A be a basic k-algebra of ﬁnite global dimension. Let n be the rank of its Grothendieck group. The Euler form of A is the Z-bilinear form deﬁned on Zn given by dim M, dim N =

∞ j (−1)j dimk Ext A (M, N), j=0

where M and N are A-modules. The symmetric Euler form is given by (dim M, dim N) = dim M, dim N + dim N, dim M . Deﬁnition 3.6.6. The Euler quadratic form of an algebra A is the form qA : Zn −→ Z given by qA (dim M) = dim M, dim M =

∞ j=0

j

(−1)j dimk Ext A (M, M),

where M is an A-module. P. Gabriel classiﬁed the quivers of ﬁnite type in terms of Dynkin diagrams, and shortly thereafter I.N. Bernstein, I.M. Gelfand, and V.A. Ponomarev gave a new and elegant proof of Gabriel’s theorem based on the connection between Dynkin diagrams and the quadratic form of quiver being positive deﬁnite. Let Q be a quiver without oriented cycles. Let n = |Q0 |.

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Deﬁnition 3.6.7. The Euler form (also known as the Ringel form) of a quiver Q is the bilinear form on Zn given by x, y = xi yi − xs(ρ) yt(ρ) . i∈Q0

ρ∈Q1

Deﬁnition 3.6.8. The symmetric Euler form (or the Cartan form) of the quiver Q is given by (x, y) = x, y + y, x . Note that the symmetric Euler form is independent of the orientation of Q. Deﬁnition 3.6.9. The Tits form of a quiver Q is deﬁned by 1 xi2 − xs(ρ) xt(ρ) . qQ (x) = x, x = (x, x) = 2 i∈Q0

ρ∈Q1

Lemma 3.6.10. Let Q be a quiver without oriented cycles. Then, the Euler quadratic form qA of the path algebra A = kQ and the Tits form of the quiver Q coincide. Moreover, xi2 − nij xi xj , qA (x) = i∈Q0

i,j∈Q0

where nij = dimk Ext 1A (Si , Sj ) = the number of arrows between i and j. Lemma 3.6.11. Let Q be a quiver without oriented cycles. The symmetric Euler form (, ) on Zn of the quiver Q (or kQ) can be given by −nij if i = j; (αi , αj ) = 2 if i = j, where αi is the ith standard coordinate vector (or the dimension vector of simple kQ-module Si ). Deﬁne C = (aij )1≤i,j≤n , where aij = (αi , αj ). Then, C is a symmetric Cartan matrix. Indeed, it is quite convenient to describe a ﬁnite quiver Q with no oriented cycles and vertex set Q0 = {1, 2, . . . , n} in terms of a Cartan matrix C = (aij ). By the above, aii = 2 for 1 ≤ i ≤ n, and −aij (= −aji ) is the number of arrows between i and j. The matrix C will then be called the Cartan matrix of a quiver Q. 1

Example 3.6.12. Consider the quiver Q : • The associated Cartan matrix of Q is ⎡ ⎤ 2 −1 0 ⎣ −1 2 −1 ⎦. 0 −1 2

- •2

- •3 .

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If Q does not contain oriented cycles, then = kQ is hereditary. The Euler form is given by dim M, dim N = dimk Hom (M, N) − dimk Ext 1 (M, N) for any –modules M and N. Since is hereditary, dim M, dim N depends only on the dimension vectors dim M and dim N. Corollary 3.6.13. Let Q be a quiver with no oriented cycles. The index set I (= Q0 ) and the symmetric Euler form (, ) of Q give a realization of a Cartan datum . Using the Tits form of a quiver, the set of roots can be characterized as follows. Deﬁnition 3.6.14. Deﬁne = {α ∈ Zn | α = 0, qQ (α) ≤ 1}, the set of roots. A root α is real if qQ (α) = 1 and imaginary if qQ (α) ≤ 0.

4. Dimension vectors and positive roots In this section, let k be any ﬁeld. We will see a close connection between the representation theory of quivers and Lie algebras. 4.1. Gabriel theorem Let g(C) be the Kac–Moody algebra associated to the Cartan matrix of a quiver Q. Sometimes, we will say that the Kac–Moody algebra g(C) and the path algebra kQ share the same Cartan datum (or the related diagram). Ignoring the orientation of Q, the underlying graph is the Dynkin graph of g(C). Based on the root space decomposition of g(C), we can obtain the root system and a set of simple roots {αi | i ∈ I} indexed by the vertices of Dynkin graph, which coincides with the root system deﬁned in 3.6.14. The following result is one of the founding stones of the representation theory of algebras. Theorem 4.1.1 [1,13]. Let Q be a quiver without oriented cycles and = kQ be the path algebra of Q. Let PInd be the set of isomorphism classes of all indecomposable -modules. If and g(C) share the same Dynkin (or Euclidean) diagram, then there exists a surjective map from PInd to + . The map is given by (dimk Mei )αi . [M] −→ i∈Q0

Moreover, if Q is of ﬁnite type, then the above map is bijective.

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Remark 4.1.2. i

• If we identify αi = dim Si = (0, 0, . . . , 1, . . . , 0, 0), then dim M = (dimk Mei )αi . i∈Q0

Thus, the dimension vector of the indecomposable -module is nothing but a (corresponding) positive root; • If there is an indecomposable representation of dimension vector α, then α is a root; • If α ∈ + real , then there is a unique indecomposable of dimension vector α (up to isomorphism); • If α ∈ + img , then there are many indecomposables of dimension vector α (up to isomorphism). In the following, we will not distinguish between the dimension vector of an indecomposable -module and the corresponding positive root of g(C). Remark 4.1.3. Gabriel’s result was generalized to arbitrary quivers in [17] by V. Kac who proved that indecomposable representations occur in dimension vectors that are the roots of Kac–Moody Lie algebra associated to a given graph. In particular, these dimension vectors do not depend on the orientation of the arrows in Q. Given a quiver with loops, the symmetric bilinear form is deﬁned by −nij if i = j, (αi , αj ) = 2 − #{loops at i} if i = j. One can deﬁne roots for any graph, and more generally for “species” (see [1]). Using species, all possible Dynkin diagrams can arise. 1

2

3

-• -•. Example 4.1.4. Consider the quiver Q: • The Auslander–Reiten quiver of the path algebra kQ looks as follows: k

0

0

0

k

k

k

k

k

k

0

k

0

k

0

k

0

0.

According to Gabriel theorem, all positive roots of a simple Lie algebra of type A3 can be listed in a manner that is compatible with the above Auslander–Reiten quiver.

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535

1

2

The positive roots of the Lie algebra of type A3 : •

3

•

• are

. 1 3 - •2 Example 4.1.5. Consider the quiver Q: . • •. The Auslander–Reiten quiver of the path algebra kQ looks as follows:

k

0

k

k

k

k

0

k

0

k

k

k

0

k

0

0 .

k

k

This Auslander–Reiten quiver has a different shape as from previous one even though the underlying graph of the above two quivers are the same. But at the dimension vector level, they are the same. All dimension vectors are the positive roots of Lie algebra of type A3 .

4.2. BGP reﬂection functors Let Q be a quiver without oriented cycles and t ∈ Q0 . Let σt Q be the quiver obtained from Q by changing the orientation of all arrows that have t as a starting or terminating vertex. Assume t is a sink for Q. We now deﬁne the Bernstein–Gelfand–Ponomarev reﬂection functor: σt+ : Rep(Q) → Rep(σt Q). Let V = (Vi , Vρ )i∈Q0 ,ρ∈Q1 be a representation of Q. The representation σt+ V = (Vi , Vρ )i∈(σi Q)0 ,ρ∈(σt Q)1

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of σt Q is deﬁned as follows: • Vi = Vi for i = t, Vt is the kernel of Vs(α) → Vt , α:s(α)→t

where the sum is taken over all arrows in Q with terminating vertex t; • Vρ = Vρ for all arrows ρ : i → j in Q1 with j = t. If ρ : i → t is an arrow in Q, then ρ : t → i is an arrow in σt Q, and Vρ is the composition of the inclusion Vs(α) with the projection onto the direct summand Vi . of Vt into α:s(α)→t

Dually, if t is a source for Q, we can deﬁne another BGP-reﬂection functor σt− : Rep(Q) → Rep(σt Q). These functors are the analog of simple reﬂections in Lie algebra theory and will play a similar role. Proposition 4.2.1. The functors σt+ and σt− induce quasi-inverse equivalences between the full subcategory of Rep(Q) of the representations having no direct summand isomorphic to St and the full subcategory of Rep(σt Q) of the representations having no direct summand isomorphic to St . Deﬁnition 4.2.2. A sequence im , . . . , i1 is called a sink sequence for Q if im is a sink for Q, and for any 1 ≤ t < m, the vertex it is a sink for the quiver σit+1 . . . σm Q. Deﬁnition 4.2.3. Let Q be a quiver with no oriented cycles and let in , . . . , i1 be a sink sequence that uses each vertex of Q exactly once. We can then deﬁne the Coxeter functors C+ = σi+1 . . . σi+n and C− = σi−n . . . σi−1 . Both are endofunctors of Rep(Q). Remark 4.2.4. Given a Cartan matrix C, one has the following correspondence between Lie theory and the representation theory of ﬁnite-dimensional algebras: Symmetric Kac–Moody algebra Symmetric Cartan matrix Root Root system Reﬂection si

↔ ↔ ↔ ↔ ↔

Hereditary path algebra Symmetric Euler form Dimension vector Auslander–Reiten quiver BGP reﬂection functor σi

5. Ringel–Hall algebras In this section, we let k be the ﬁnite ﬁeld Fpr . Let Q be a quiver without oriented cycles, and let = kQ be the corresponding path algebra that is a ﬁnite-dimensional hereditary algebra over the ﬁnite ﬁeld k. Recall that the index set I = Q0 and the symmetric Euler form of Q together give a realization of a Cartan datum .

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5.1. Ringel–Hall algebras Let P be the set of isomorphism classes of ﬁnite-dimensional -modules and I ⊂ P be the set of isomorphism classes of simple -modules. We choose a representative Vα ∈ α for any α ∈ P. For α, β ∈ P, we write α, β = dim Vα , dim Vβ , and that deﬁnes the Euler form −, − on ZI . The symmetric Euler form (−, −) is given by (α, β) = α, β + β, α on ZI . Let R be a commutative integral domain containing Q(v), where v2 = q, q = |k|, and Q(v) is the rational function ﬁeld of v. Deﬁnition 5.1.1 (C. M. Ringel). The Ringel–Hall algebra H() is the free R-module with • basis: {uα | α ∈ P} = {isomorphism classes of f. d. − modules}; • multiplication given by λ gαβ uλ uα uβ = vα,β λ∈P

λ is the number of submodules V of V such that for all α, β ∈ P, where gαβ λ λ = Vλ /V and V lie in the isomorphism classes α and β respectively, that is, gαβ #{V ⊆ Vλ | V Vβ , V/V Vα }.

It is easy to verify that H() is an associative NI -graded R-algebra with the identity element u0 . The grading H() = ⊕r∈NI Hr is deﬁned as follows: for each r ∈ NI, Hr is the R-span of the set {uλ | λ ∈ P, dim Vλ = r}. We denote by C() the R-subalgebra of H(), which is generated by ui , i ∈ I. C() is called the composition algebra of . Lemma 5.1.2. If is of ﬁnite type, then H() = C(). Remark 5.1.3. If is of ﬁnite type, whenever we change the ground ﬁeld to be λ can be evaluated by a polynomial at |k|, another ﬁnite ﬁeld, the ﬁltration number gαβ λ λ λ . The polynomial φλ occurring in that is, gαβ = φαβ (|k|) for some polynomial φαβ αβ this way is called a Hall polynomial. C.M. Ringel [18, 19] has proved that the elements ui , i ∈ I satisfy the quantum Serre relations 1−aij 1−a −t t 1 − aij (−1) uti uj ui ij = 0, t t=0

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for any i = j in I, where aij =

2(i, j) = (i, j) = (dim Si , dim Sj ), and (i, i)

vs − v−s = vs−1 + vs−3 + · · · + v−s+1 v − v−1 s [s]! s [s]! = [r] , = r [r]![s − r]! [s] =

r=1

where r, s are non-negative integers, and r ≤ s. 5.2. Quantum groups and Ringel–Hall algebras Let k¯ be the algebraic closure of k. For any n ∈ N\{0}, let F(n) be a subﬁeld of k¯ such that [F(n) : k] = n. If we let (n) = ⊗k F(n), then (n) is a ﬁnite-dimensional hereditary F(n)-algebra corresponding to the same Cartan datum as that of . We also have the Hall algebra Hn = Hn ((n)) of (n). Deﬁne H= Hn . n>0

Without √ creating confusion, we still use letter v to represent v = (vn )n ∈ H, where vn = |F(n)|. Obviously, v lies in the center of H and is transcendental over the rational ﬁeld Q. Let ui = (ui (n))n ∈ H satisfy that ui (n) is the element of H((n)) corresponding to Si (n), where Si (n) is the simple (n)-module that lies in the class i. Deﬁnition 5.2.1. The generic composition algebra C () of the Cartan datum is deﬁned to be the subring of H generated by Q and the elements v, v−1 , and ui (i ∈ I). The integral generic composition algebra A C () is the A–subalgebra of um (m) C () generated by ui = i (i ∈ I), where A = Z[v, v−1 ]. [m]! Let Uv+ (g) be the positive part of the Drinfel’d–Jimbo quantum group corresponding to the Cartan datum . A fundamental theorem of Green and Ringel establishes a wonderful connection between generic composition algebras and quantum groups. Theorem 5.2.2 [18, 20]. If g and have the same Cartan datum, then there is a Q(v)-algebra isomorphism C () Uv+ (g), which maps ui −→ Ei for all i ∈ I. Even more, A C () is isomorphic to A Uv+ (g) (m) (m) by sending ui to Ei for all i ∈ I. Thanks to this isomorphism, our discussion is applicable directly to generic composition algebras and quantum groups, even though we have started with our considerations with a composition algebra over ﬁnite ﬁeld.

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539

Remark 5.2.3. If we extend Uv+ (g) to the Borel part of Uv (g) and C () to a certain extension C(, T), then η can be extended canonically to be a bijection of Hopf algebras (see [20–22]). Remark 5.2.4. If is of ﬁnite type, the existence of Hall polynomials makes it possible to deﬁne the generic composition algebra by λ uα uβ = vα,β φαβ (v2 )uλ , λ∈P

where v is an indeterminate. So far, all the Kac–Moody algebras considered have been symmetric. For nonsymmetric cases, one should use “species” (introduced by V. Dlab and C. M. Ringel in [1]) instead of quivers to construct an algebra that realizes the positive part of the corresponding quantum group. 6. PBW basis and canonical basis Let g = g(C) be the Kac–Moody algebra of type C, let U = Uv (g) be the corresponding quantum group, and let U+ = Uv+ (g) be its positive part. For simplicity, we only consider the case when the Cartan matrix C is symmetric. Let Q be a quiver without oriented cycles whose Cartan matrix is C, and let = kQ be the hereditary path algebra of Q. We will not consider the quivers An with cyclic n ) consisting of orientation in this chapter. Nevertheless, the subcategory of Rep(A all nilpotent representations of the cyclic quiver An plays a very important role in the representation theory of algebras. Because of the canonical isomorphism between U+ and the generic composition algebra C () of (where g and have the same diagram), we are going to be able to use representation theory of algebras to analyze the structure of the corresponding Kac–Moody algebras and related quantum groups. The so-called canonical basis of U+ was ﬁrst introduced by G. Lusztig in [4] via an elementary algebraic deﬁnition in the case when g is simple ﬁnite-dimensional Lie algebra. Subsequently, G. Lusztig studied canonical bases in the framework of Ringel–Hall algebras. The canonical basis was characterized by being integral, being invariant under the bar involution, and spanning a certain Z[v−1 ]-lattice L with a speciﬁc image in the quotient L/v−1 L, where L is generated by any basis of PBW type. This algebraic deﬁnition does not work for quantum group of arbitrary Kac–Moody algebras. The difﬁculty in constructing a basis for L arises from the need to deﬁne suitable analogs of imaginary root vectors. By using topological methods, G. Lusztig [5] constructed the canonical basis of a quantum group of symmetric Kac–Moody algebra. It was shown that the “global crystal basis” introduced by M. Kashiwara and the canonical basis coincide. The canonical basis has very nice properties, but there are also still many unanswered questions.

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There are different approaches to the construction of the canonical basis: an algebraic approach [4, 23–25], a geometric approach [5], and a quiver approach [26, 27]. We will not comment on the geometric approach here. To construct the canonical basis, we start with a PBW basis, which involves root vectors. 6.1. Root vectors in quantum groups According to G. Lusztig [28], there exists an action of the braid group corresponding , T − , i ∈ I of the braid group to to on U. Applying the standard generators Ti,1 i,1 the generators of U in an admissible order, we obtain a family of linearly independent elements in U+ . Since these elements degenerate into a basis of ⊕α∈+ gα by the real specialization v → 1, we call these elements the real root vectors of U+ . If is of ﬁnite type, they provide a complete set of root vectors. Note that by using results of C. M. Ringel [27] and J. Xiao [22], the braid group action can be realized through BGP-reﬂection functors. The operations Ti,1 are given by σi on the Ringel–Hall algebra according to the canonical isomorphism C () U+ . For afﬁne Kac–Moody algebra g, there exist imaginary roots. Several authors have introduced imaginary root vectors for U (see [23, 29–31]). Those imaginary root vectors cannot be obtained by Lusztig’s operations. Since the Auslander–Reiten quiver is a convenient device for visualizing the module category of a ﬁnite-dimensional algebra, we may ask what kind of information about root vectors in U+ can be read off from the Auslander–Reiten quiver of . C. M. Ringel [27] gives an answer, providing an interpretation for the root vectors obtained by Lusztig’s braid group action in C (). 6.2. PBW basis and canonical basis for C () of ﬁnite type We start with a path algebra = kQ, where k is a ﬁnite ﬁeld and Q is of ﬁnite type. Let g be a ﬁnite-dimensional simple Lie algebra of type A, D, or E, and let + be the set of all positive roots, W its Weyl group, and U the corresponding quantum group. Since is of ﬁnite type, there are ﬁnitely many isomorphism classes of indecomposable modules. The isomorphism classes of all indecomposables are in one-to-one correspondence with the positive roots + of the corresponding simple Lie algebra. Every indecomposable –module can be uniquely determined by a positive root α ∈ + . We will denote this indecomposable -module by Vα if dim Vα = α. It is well known that mod − (identiﬁed with Rep(Q)) is representation-directed, (see [1, 32]); this means that there exists a total ordering of the positive roots, say β1 , β2 , . . . , βN , where N is the cardinality of + such that Hom(Vβi , Vβj ) = 0 implies that i ≤ j. Such an ordering is called Q-admissible. Consider this Q-admissible ordering of positive roots. There exists a sequence i1 , . . . , iN of vertices of Q, with the following properties: (1) the sequence iN , . . . , i1 is a sink sequence for Q; (2) we have βl = si1 si2 . . . sil−1 (αil ), where αil is a simple root.

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There exists a reduced expression of the longest element w0 = si1 . . . siN−1 siN in W such that iN , iN−1 , . . . , i1 is a sink sequence for Q derived from an Q-admissible ordering of positive roots. The element w0 gives a total ordering on + , which is the same as the Q-admissible ordering: β1 = αi1 < β2 = si1 (αi2 ) < · · · < βN = si1 si2 · · · siN−1 (αiN ). G. Lusztig deﬁned the root vectors in U by Eβl := Ti1,1 Ti2,1 . . . Til−1 ,1 (Eαil ).

Theorem 6.2.1 [27]. In C (), Eβl = vθ uβl , where θ = −dimk (Vβl ) + dimk End Vβl . +

Let N be the set of multiplicity functions from + = {β1 , β2 , . . . , βN } to N. + Given c ∈ N , we deﬁne (c(β1 ))

Ec = Eβ1 (r)

where Eβi =

Eβr i

[r]!

(c(β2 ))

Eβ2

(c(βN ))

. . . EβN

,

for any r ∈ N. +

Theorem 6.2.2 [4, 27]. {Ec | c ∈ N } is a PBW A-basis of A U+ . This PBW basis depends on the choice of a reduced expression of the longest element w0 . +

Theorem 6.2.3 [4, 24]. For each given c ∈ N , there exists a unique b(c) ∈A U+ such that (a) b(c) = b(c); (b) b(c) = Ec + c
The basis {b(c) | c ∈ N } is the canonical basis introduced by G. Lusztig. This basis is independent of the choice of the reduced expression of the longest element w0 . A geometric realization of the canonical basis can be found in [4], where the theory of perverse sheaves is involved. 6.3. Root vectors in C () of tame type We start with a path algebra = kQ where k is a ﬁnite ﬁeld and Q is of tame type. or Let g be a symmetric afﬁne Kac–Moody algebra of type A, D E, let W be its Weyl group and U be the corresponding quantum group.

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To construct a PBW basis and a canonical basis of U, some new problems arise, which do not appear in the ﬁnite case (cf. [30]): (1) there do exist imaginary roots mδ (m ∈ Z\{0}), which are ﬁxed by the action of Weyl group, so we cannot apply the action of braid group to the simple root vectors to obtain the imaginary root vectors of a quantum group; (2) there does no longer exist a longest element of the Weyl group. In the ﬁnite case, the longest element w0 gives a way to totally order all positive roots. We need to ﬁnd another way to order all roots; (3) the real roots are divided into two disjoint orbits by the action of the Weyl group. Let I = {0, 1, 2, . . . , n} be the index set of all simple roots of g. Deﬁne a doubly inﬁnite sequence . . . i−1 , i0 , i1 , i2 . . . by setting il = il(mod N) , where N is the length of some special element in W. For a detailed description of this special element, we refer to [23, 24]. Note that sim . . . sit is reduced for any m < t. Set βl =

si0 si−1 . . . sil+1 (αil )

if l ≤ 0,

si1 si2 . . . sil−1 (αil )

if l > 0.

Lemma 6.3.1 [23, 24]. {βl | l ∈ Z} = + real . The imaginary roots cannot be obtained in the same manner (through the action of Weyl group on simple roots). Let δ be the smallest imaginary root. Then the imaginary roots of g are the elements lδ for l ∈ Z, l = 0 and mult (lδ) = n. Let + img = {(lδ, t) | l > 0, 1 ≤ t ≤ n} be the set of all imaginary roots counted with multiplicity. Deﬁne a total order on + by setting β0 < β−1 < · · · < (δ, 1) < · · · < (δ, n) < (2δ, 1) < · · · (2δ, n) < · · · < β2 < β1 . We now deﬁne real root vectors for each positive real root as follows: Eβl =

−1 −1 T −1 i0 ,1 T i−1 ,1 . . . T il+1 ,1 (Eαil ) Ti1,1 Ti2,1 . . . Til−1 ,1 (Eαil )

if l ≤ 0, if l > 0.

Having deﬁned real root vectors, we continue to deﬁne the imaginary root vectors. For l > 0, i ∈ I \{0}, set ψlδ,i = Elδ−αi Eαi − v−2 Eαi Elδ−αi .

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543

Following [29], we introduce “integral” imaginary root vectors Elδ,i (l ≥ 0, 1 ≤ i ≤ n) in U+ by E0,i = 1 and l 1 t−l v ψtδ,i E(l−t)δ,i . Elδ,i = [l] t=1

Let Q be a quiver with no oriented cycles and |Q0 | = n + 1. Let in+1 , in , . . . , i1 be a sink sequence that uses each vertex of Q exactly once. We can deﬁne the Coxeter functors: C+ = σi+1 . . . σi+n+1 and C− = σi−n+1 . . . σi−1 . Both are endofunctors of mod − (or Rep(Q)). According to Theorem 4.1.1, if α ∈ + real , then there is a unique indecomposable –module of dimension vector α (up to isomorphism) and if α ∈ + img , then there are many indecomposable –modules of dimension vector α (up to isomorphism). We can use the Coxeter functors C± to partition all indecomposable -modules. Deﬁnition 6.3.2. Given an indecomposable -module V , we call it • preprojective indecomposable module if (C+ )r V = 0 for r 0; • preinjective indecomposable module if (C− )r V = 0 for r 0; • regular indecomposable module if it is neither preprojective nor preinjective. Moreover, (C± )r V V for some r > 0. All preprojective and preinjective modules may be constructed in a uniform way through the action of BGP reﬂection functors. For 1 ≤ l ≤ n + 1, we can obtain all indecomposable projective -modules by Pil = σi−n+1 σi−n σi−l+1 (Sil ), where Sil is the simple module of the path algebra of σil σil−1 . . . σi1 Q. Dually, all indecomposable injective -modules are given by Iil = σi+1 σi+2 σi+l−1 (Sil ), where Sil is the simple module of the path algebra of σil σil+1 . . . σin+1 Q. If V is an indecomposable preprojective module, then

t

σi−n+1 σi−n σi−l+1 (Sil ) V (C− )t (Pil ) = σi−n+1 . . . σi−1 for some t ≥ 0. Even more, dim V = (sin+1 . . . si1 )r (sin+1 sin sil+1 (αil )). If V is an indecomposable preinjective module, then

t

σi+1 σi+2 σi+l−1 (Sil ) V (C+ )t (Iil ) = σi+1 . . . σi+n+1 for some t ≥ 0. Also dim V = (si1 . . . sin+1 )r (si1 si2 sil−1 (αil )). Using a sink sequence in+1 , in , . . . , i1 of all vertices of Q, we can deﬁne a doubly inﬁnite sequence . . . , i−1 , i0 , i1 , i2 , . . . by setting il = il(mod n+1) for l ∈ Z.

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Set βˆ l =

si0 si−1 . . . sil+1 (αil )

if l ≤ 0,

si1 si2 . . . sil−1 (αil )

if l > 0.

We have already shown that βˆ l for l ≤ 0 is the dimension vector of an indecomposable preprojective module and βˆ l for l > 0 is the dimension vector of an indecomposable preinjective module. Using the actions of BGP reﬂections on simple modules, we can only get all the indecomposable preprojective and preinjective modules. According to the Auslander–Reiten quiver of , the set + can be partitioned as follows. The roots {βˆ l | l ∈ Z} are called irregular roots. The dimension vector of any indecomposable regular module is called a regular root. Hence, we have + = {irregular roots} {regular roots}. The Auslander–Reiten quiver of looks as follows:

There exists a unique preprojective (or preinjective) component that consists of all indecomposable preprojective (or preinjective) modules. There are lots of regular components containing all indecomposable regular modules. We have the following geometric picture of a regular component of the AR–quiver:

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Here there is the possibility that (C+ )n V0 V0 for some n > 0. Then (C+ )n Vi Vi for all i and we can identify all points on those two vertical lines to obtain what is called a stable tube. If n = 1, the tube is called homogeneous stable tube. Note that only positive real root uniquely determines an indecomposable -module (up to isomorphism). + Lemma 6.3.3 [27]. {βˆ l | l ∈ Z} ⊆ + real and {Regular roots} = img {Regular real roots}.

We can order all irregular roots as βˆ 0 < βˆ −1 < βˆ −2 < · · · < βˆ 3 < βˆ 2 < βˆ 1 . Let Vβˆ l for l ∈ Z be the indecomposable preprojective or preinjective module with dim Vβˆ l = βˆ l . The above order of positive roots is compatible with the structure of the preprojective and preinjective components of the Auslander–Reiten quiver of , which means Hom (Vβˆ j , Vβˆ t ) = 0 implies βˆ j < βˆ t . Similarly as before, we can deﬁne real root vectors for each irregular real root: ⎧ ⎨ Ti,1 −1 Ti ,1 −1 . . . Ti ,1 −1 (Eαi ) if l ≤ 0, l 0 −1 l+1 Eβˆ l = ⎩ T T . . . T (Eα ) if l > 0. il i1 ,1 i2 ,1 il−1 ,1 can be realized by σ under the identiﬁcation C () U+ . Recall that Ti,1 i

Theorem 6.3.4 [27]. Let Vβˆ l be an indecomposable preprojective or preinjective -module with dim V ˆ = βˆ l for l ∈ Z, then in C () (or U+ ), βl

Eβˆ l = vθ uβˆ l , where θ = −dimk (Vβˆ l ) + dimk End Vβˆ l . Proposition 6.3.5 [33, 34]. There is an algorithm to express Eβˆ l as linear combination of simple root vectors. In this way, we only construct irregular real root vectors. Now the natural next question is how to construct imaginary and regular real root vectors in Ringel–Hall 1 —the Kronecker quiver, we answer the question algebras. For the quiver of type A in next subsection. The question still remains open for the general case. 6.4. Kronecker quiver In this section, we only consider the Kronecker quiver Q : 2 • - • 1. + $2 . Let k be a ﬁnite ﬁeld. The Let be the positive root system of Lie algebra sl path algebra K = kQ is the “smallest” tame hereditary algebra. We will provide a

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realization of all imaginary root vectors in the generic composition algebra C (K). Moreover, an integral PBW basis and the canonical basis of this algebra are obtained. $2 ). The corresponding quantum group is Uv (sl The Auslander–Reiten quiver of K consists of one preprojective component, one preinjective component (which have the same structure as those components of kQ for k being algebraically closed ﬁeld), and a family of homogeneous tubes of regular modules parameterized by the set of all monic irreducible polynomials over k (which is different from that of kQ when k is an algebraically closed ﬁeld). Let α1 = dim S1 and α2 = dim S2 . Put δ = α1 + α2 . The indecomposable preprojective and preinjective modules have the dimension vectors nδ + α1 and nδ + α2 with n ∈ N, respectively. Moreover, each indecomposable preprojective and preinjective module is uniquely determined by its dimension vector. The indecomposable regular modules have the dimension vectors nδ with n ∈ N \ {0}. There is no regular real root. Every regular root is an imaginary root. Put (r, t) = rα1 + tα2 . We have + = {(l + 1, l), (m, m), (n, n + 1) | l ≥ 0, m ≥ 1, n ≥ 0}, which is the set of dimension vector of all indecomposable representations of the Kronecker quiver. The Auslander–Reiten quiver of K looks as follows:

In U+ , we have for i = j ∈ {1, 2} Enδ+α1 = T1,1 %

Enδ+α2 =

−1

T2,1

−1

T1,1 &'

−1

. . . Ti,1

n times

−1

(Ej ), (

T2,1 T1,1 T2,1 . . . Ti,1 (Ej ),

%

&'

n times

(

and T are the braid group operations deﬁned in [28]. where T1,1 2,1

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547

According to C. M. Ringel [27], we can deﬁne the following root vectors in C (K), which correspond to all positive real roots: E1 = Eα1 = u1 ,

E2 = Eα2 = u2 ,

Enδ+α1 = v−dimk Vnδ+α1+dimk End (Vnδ+α1 ) unδ+α1 = v−2n unδ+α1 , Enδ+α2 = v−dimk Vnδ+α2+dimk End (Vnδ+α2 ) unδ+α2 = v−2n unδ+α2 . for any n ∈ N. Following I. Damiani [30] (or [23]), there are the imaginary root vectors ψnδ = Enδ−α1 E1 − v−2 E1 Enδ−α1 ,

for n ∈ N\{0},

and integral imaginary root vectors n

E0δ = 1 and Enδ =

1 i−n v ψiδ E(n−i)δ . [n] i=1

Theorem 6.4.1 [35, 36]. In C (K), we have Enδ = v−2n uα . α ∈ P, Vα regular, dimVα = nδ

The total ordering on + (adapted for the structure of the Auslander–Reiten quiver) is as follows: α1 < δ + α1 < · · · < (n − 1)δ + α1 < nδ + α1 < · · · · · · < δ < 2δ < · · · < (n − 1)δ < nδ < · · · < nδ + α2 < (n − 1)δ + α2 < · · · < δ + α2 < α2 . +

Let N be the set of ﬁnitely supported multiplicity functions from + to N. + Given c ∈ N , if {α ∈ + | c(α) = 0} = {β1 < β2 < . . . βn }, deﬁne Ec be the monomial formed by multiplying the divided powers of real root vectors and imaginary root vectors, that is, (c(β1 ))

Ec = Eβ1 (c(β )) Eβm m

c(β1 )

Eβ1

(c(β2 ))

Eβ2

(c(βn ))

. . . Eβn

, (c(βm ))

if βm is real, and Eβm [m]! Then, there is the following theorem:

where

=

+

c(β )

= Eβm m if βm is imaginary.

Theorem 6.4.2 [35, 37]. The set {Ec | c ∈ N } is a PBW A–basis of A C (K) (or + $ A Uv (sl2 )).

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X. Chen, K.–B. Nam and T. Pospíchal (d )

(d )

+

For d = (d1 , d2 ) ∈ N2 , write E(d) = E2 2 E1 1 . Similarly, for c ∈ N , deﬁne E(c) = E(c(β1 ))E(c(β2 )) . . . E(c(βm )), which is a monomial on the divided powers of the Chevalley generators E1 and E2 . Therefore, E(c) ∈ C (K). We will now describe a triangular relation between the PBW basis and the above monomial basis. + Let ϕ : N → N2 be the map given by ϕ(c) = c(α)α. α∈+

Then for any d ∈ N2 , ϕ−1 (d) gives the size of all isomorphism classes of modules, which have the dimension vector d and is a ﬁnite set. +

Proposition 6.4.3 [26]. For any c ∈ N , E(c) = hcc Ec c ∈ϕ−1 (ϕ(c))

such that (1) hcc ∈ Z[v, v−1 ]; (2) hcc = 1; (3) if hcc = 0 and c = c, then c < c, where < is the geometric order related to the corresponding modules; (4) E(c) is bar-invariant, that is, E(c) = E(c). +

Proposition 6.4.4 [26]. For any c ∈ N , set Ec = ωcc Ec . +

c ∈N

Then, (1) ωcc ∈ Z[v, v−1 ]; (2) ωcc = 1; (3) if ωcc = 0 and c = c, then c < c. +

Consider the bar involution in Section 2, we have Ec = Ec for any c ∈ N . Hence, Ec =

c

ωcc Ec =

c ,c

ωcc ωcc Ec .

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549

+

Since {Ec | c ∈ N } is an A–basis of C (K), we have ωcc ωcc = δcc , c

and therefore, one can uniquely solve the system of equations ωcc ζcc ζcc = c ≤c ≤c

with unknowns ζcc ∈ Z[v−1 ] such that (1) ζcc = 1; (2) ζcc ∈ v−1 Z[v−1 ] for all c < c. For any c ∈ ϕ−1 (d) and d ∈ N2 , set εc = ζcc Ec . c ∈ϕ−1 (d)

Theorem 6.4.5 [26]. The set {εc | c ∈ ϕ−1 (d), d ∈ N2 } provides the canonical basis of A C (K). The importance of the Kronecker quiver lies in the existence of a full exact embedding from the category of regular modules of K to the category of regular modules n , of any tame hereditary algebra with underlining quiver An , D E6 , E7 , or E8 . Theorem 6.4.6 [1]. There exists a fully faithful exact functor F : mod − K −→ mod − kQ, which preserves regular modules, where Q = An (excluding the case of a cyclic n , E6 , E7 , or E8 . Even more, this functor gives rise to an injection of orientation), D Ringel–Hall algebras (which will still be denoted by F). More recently, P. Baumann and C. Kassel [38] described the Ringel–Hall algebra of the category of coherent sheaves on the projective line and recovered M. Kapranov’s isomorphism between a certain subalgebra of this Ringel–Hall algebra and the “pos$2 ). In combination with the isomorphism between the generic itive part” of Uv (sl $2 ), the real root vectors Enδ+α and the imagcomposition algebra C (K) and Uv+ (sl 1 inary root vectors Enδ are related to locally free coherent sheaves and the torsion sheaves, respectively. 6.5. PBW basis and canonical basis for C () of tame type We use the same assumptions presented in Subsection 6.3. There are different approaches to the construction of the canonical basis of U+ . G. Lusztig [5]

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X. Chen, K.–B. Nam and T. Pospíchal

parameterized the canonical basis elements by using the representation theory of Q and algebraic geometry. J. Beck, V. Chari, and A. Pressley [23] provided an algebraic characterization of the canonical basis. J. Beck and H. Nakajima [24] obtained a purely algebraic construction of the PBW basis. Z. Lin, J. Xiao, and G. Zhang [26] used the Ringel–Hall algebra approach to construct the PBW basis and the canonical basis of C (). + + prep be the set of ﬁnitely ˆ ˆ Let + prep = {βl | l ≤ 0} and prei = {βl | l > 0}. Let N supported multiplicity functions from + prep to N. Similarly, we deﬁne N Given c+ ∈ N

+ prep

+ prei

.

, if

ˆ ˆ ˆ {α ∈ + prep | c+ (α) = 0} = {β1 < β2 < · · · < βn }, deﬁne Ec+ to be the monomial formed by (c(βˆ 1 ))

E c+ = E ˆ

β1

)) (c(βˆ m βm

where E ˆ

=

E

(c(βˆ 2 ))

Eˆ

β2

) c(βˆ m ˆβm

(c(βˆ n )) , βn

· · · Eˆ

. Similarly, deﬁne Ec− for c− ∈ N

+ prei

. [m]! We now deﬁne a set M = {(c+ , (π1 , π2 , . . . , πs ), w, c− )} by the following rule: • (π1 , π2 , . . . , πs ) is a collection of partitions and each πt is related to a nonhomogeneous tube; • w is a partition (w1 , w2 , . . . , wt ) for some t.

Then for each c ∈ M, deﬁne Ec = Ec+ Eπ1 Eπ2 · · · Eπs F(Ew1 δ )F(Ew2 δ ) · · · F(Ewt δ )Ec+ , where Eπ1 Eπ2 · · · Eπs is the product of some elements in C (), which are related to all nonhomogeneous tubes, and F is the functor mentioned in 6.4.6. We omit the detailed discussion; interested readers are referred to [26]. Theorem 6.5.1 [26]. The set {Ec | c ∈ M} is an A-basis of A C (). The set {εc | c ∈ M} provides the canonical basis of A C (), where εc is deﬁned in a similar manner as was done for the Kronecker quiver.

7. Root categories, Kac–Moody algebras and elliptic Lie algebras 7.1. A construction of the Kac–Moody algebra g (C). Let be a ﬁnite-dimensional hereditary algebra over a ﬁeld k (or, more speciﬁcally, = kQ for some quiver Q without oriented cycles). Let g(C) be a symmetrizable Kac–Moody algebra associated to a symmetrizable Cartan matrix C, and let g (C) be its derived algebra, that is, g (C) = [g(C), g(C)]. According to Theorem 4.1.1, if and g(C) (or g (C)) share the same Cartan datum, then the map dim induces a surjection between the isomorphism classes of all

Quivers and Representations

551

indecomposable –modules and the positive roots of g(C) (or g (C)). Moreover, if is of ﬁnite type, the map dim is bijective. It is natural to ask whether it is possible to recover the Lie algebra structure of g (C) in terms of –modules. For of ﬁnite type, this was successfully done by C. M. Ringel [39,40] via the Ringel–Hall algebra construction. Let K(mod −) be the free Abelian group with basis indexed by the set of isomorphism classes of indecomposable –modules. C.M. Ringel ) showed+that K(mod −) is the Chevalley Z-form of n+ and hence K(mod −) Z C and n are isomorphic Lie algebras, where g (C) = n− h n+ is a triangular decomposition of g (C). One of the important contributions to the representation theory of algebras is the study of derived categories by D. Happel [41, 42]. The derived category Db () is a triangulated category with the translation T obtained from the category of bounded complexes over by localizing with respect to the set of all quasi-isomorphisms. The interested readers are referred to B. Keller’s paper [43] in an earlier volume of this handbook for a detailed description of derived categories and their uses. There is a full embedding of mod − into Db () as a full subcategory, which sends each -module X into the stalk complex X• = (Xi , d i ) with X0 = X, Xi = 0 for i = 0, and d i = 0 for all i ∈ Z. We keep using the same symbol mod − for this subcategory. The dimension vectors of all indecomposable complexes in mod − are all positive roots and the dimension vectors of all indecomposable complexes in T (mod −) are all negative roots. 1 - •2 - •3 . The categorical Example 7.1.1. Consider the quiver Q : • structure of the derived category Db () can be visualized as follows:

,

where each indecomposable object is represented by its dimension vector. To recover the whole Lie algebra g (C) (not only n+ ) using modules of algebras, we should consider the orbit category R = Db ()/T 2 of Db () under the automorphism T 2 . The translation T of a derived category Db () induces an automorphism of R of order 2, which is again denoted by T . R inherits a triangulated category structure from Db () with T as the translation. Note that mod − can be embedded into R as a full subcategory. The root category R is divided by mod − into two parts, mod − and T (mod −). D. Happel [41] pointed out that if is of ﬁnite type, then the isomorphism classes of indecomposable objects in R correspond bijectively to all roots of g (C) (not only positive roots). For this reason, R is called the root category. It was soon found that

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X. Chen, K.–B. Nam and T. Pospíchal

the category R = Db ()/T 2 provided a solid framework, which is compatible with the root system of g (C) even when is not of ﬁnite type. In [11], V. Kac asked a fundamental question about the realization of nonafﬁne Lie algebras. This question was answered by L. Peng and J. Xiao in [2]. Not only that, the authors actually gave a realization of all symmetrizable Kac–Moody algebras in a global way via the root categories R = Db ()/T 2 . In the remainder of this section, we will summarize their work. Assume now that the ﬁeld k is a ﬁnite ﬁeld and q = |k|, the cardinality of k. Given a root category R of , we consider the Grothendieck group G(R) as usual, that is, G(R) is the quotient of a free Abelian group with basis {[M] | M ∈ R} indexed by the isomorphism classes of all objects in R, subject to the relations [Y ] − [X] − [Z] arising from the triangles of form X → Y → Z → T X in R. For any M ∈ R, we denote by dim M the canonical image of M in G(R), called the dimension vector of M. Given X, Y, L ∈ R, put W(X, Y, L) = {(f, g, h) ∈ Hom(X, L) × Hom(L, Y ) × Hom(Y, T X) | f

g

h

X → L → Y → T X is a triangle}. One can deﬁne an action of Aut(X) × Aut(Y ) on W(X, Y, L) as follows: (a, c) ◦ (f, g, h) = (fa, c−1 g, (T a)−1 hc) for any (a, c) ∈ Aut(X) × Aut(Y ), (f, g, h) ∈ W(X, Y, L). Denote the orbit space by V(X, Y, L) = W(X, Y, L)/Aut(X) × Aut(Y ). V(X, Y, L) is a ﬁnite set and L FY,X = |V(X, Y, L)|

denotes the cardinality of V(X, Y, L). Let ind R denote the set of representatives of isomorphism classes of all indecomposable objects in R. For any indecomposable object X ∈ R, let d(X) = dimk (End X/ rad End X). For any M ∈ R, write hM for the dimension vector dim M of M. Then the Grothendieck group G(R) is generated by {hM | M ∈ R} and hL = hM + hN if −hT M . Consider there exists a triangle of ) form M → L → N → T M. Thus, hM = ) the rational extension Q Z G(R). Denote by h the subgroup of Q Z G(R) generhM , M ∈ ind R. Note that the Grothendieck group G(R) is a subgroup of h. ated by d(M) Now)deﬁne a symmetric bilinear function (−|−) on h × h (and by extension also ) on (Q Z h) × (Q Z h)) by (hX |hY ) = dimk Hom(X, Y ) − dimk Hom(X, T Y ) + dimk Hom(Y, X) − dimk Hom(Y, T X) for any X, Y ∈ R. On G(R), this is just the symmetric Euler form of R.

Quivers and Representations

553

Let n be the free Abelian group with a basis {uX |X ∈ ind R}, which is bijective to the set of isomorphism classes of all indecomposable objects in R. Let g =h⊕n be a direct sum of Z-modules. Consider the quotient group g(q−1) = g/(q − 1)g. Mildly abusing notation, denote by uM , hM also the residues in the quotient group corresponding to M ∈ R. Theorem 7.1.2 [2]. g(q−1) is a Lie algebra over Z/(q − 1) with its bilinear operation [−, −] deﬁned as follows: (1) for any two indecomposable objects X, Y ∈ R, ⎧ L L ⎪ FXY if Y T X, − FYX ⎪ ⎪ ⎨ L∈ind R [uX , uY ] = ⎪ ⎪ ⎪ hX ⎩ if Y T X; d(X) (2) for any objects X, Y ∈ R with Y indecomposable, [hX , uY ] = −(hX |hY )uY and [uY , hX ] = −[hX , uY ]; (3) [h, h ] = 0, ∀ h, h ∈ h. Remark 7.1.3. Indeed, the positive part of the above Lie algebra can be obtained through the (untwisted) Ringel–Hall algebra. Recall the deﬁnition of untwisted Ringel–Hall algebra H(). It is a free Abelian group with basis {u[M] | M ∈ mod −} indexed by the isomorphism classes of all ﬁnite-dimensional –modules. Write simply uM for u[M] . The multiplication is deﬁned by M u X uY = gXY uM . [M],M∈mod −

This is an associative ring with unit u0 . Let ind be the set of representatives of isomorphism classes of all indecomposable modules in mod −, and let K(mod −) be the subgroup of H() generated (freely) by {uX | X ∈ ind }. Then the proof of Proposition 5 in [39] implies that over the residue ring Z/(q−1), K(mod −) is a Lie subalgebra of H(), that is, the quotient group K(mod −)/(q − 1)K(mod −) is a Lie subalgebra of the quotient ring H()/(q − 1)H(), where the Lie bracket is the commutator [uX , uY ] = uX uY − uY uX . The root category R has two parts, mod − and T (mod −). Hence, n = K (mod −) ⊕ K(T (mod −)), where K(T (mod −)) is the free Abelian group with

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X. Chen, K.–B. Nam and T. Pospíchal

the basis {uT (X) | X ∈ ind } and g = K(mod −) ⊕ h ⊕ K(T (mod −)). Hence, the Lie algebra K(mod −)/(q − 1)K(mod −) is a subalgebra of g(q−1) and g(q−1) = K(mod −)/(q − 1)K(mod −) ⊕ h/(q − 1)h ⊕ K(T (mod −))/(q − 1)K(T (mod −)) is a triangular decomposition of g(q−1) . Let E be a ﬁeld extension of k. Then, ⊗k E is a hereditary E-algebra, and for M ∈ mod −, the module M ⊗k E has a canonical ⊗k E-module structure. Recall that E is called conservative extension for X ∈ ind if (End / rad End X)⊗k E is a ﬁeld again so that X⊗k E is an indecomposable ⊗k E-module. An indecomposable –module X is called exceptional -module if Ext 1 (X, X) = 0. Note that all simple -modules are exceptional. It follows from C.M. Ringel [44] that for an exceptional module, the endomorphism algebra is isomorphic to the endomorphism algebra of some simple -module S. Let k¯ be the algebraic closure of k and set = {E : k ⊆ E ⊆ k¯ is a ﬁnite ﬁeld extension and conservative for all exceptional -modules} = {E : k ⊆ E ⊆ k¯ is a ﬁnite ﬁeld extension and conservative for all exceptional objects in R}. The set is an inﬁnite set. Let RE be the root category of ⊗k E. Then, as before, we have hE , nE , and E = gE /(|E|−1)gE which is a Lie algebra over Z/(|E|−1). gE = hE ⊕nE , and g(|E|−1) Consider the direct product E g(|E|−1) E∈

+ of Lie algebras and let LE(R) be the Lie subalgebra of E∈ gE (|E|−1) generated by the set {uX = (uX⊗k E )E∈ | X is an exceptional object in R}. Theorem 7.1.4 [2]. If and g (C) share the same Cartan datum, then there is a Lie algebra isomorphism : C ⊗Z LE(R) −→C g (C) given by hSi d(Si ) uSi

→

hi ,

1 ≤ i ≤ n,

→

ei ,

1 ≤ i ≤ n,

uT (Si )

→

−fi ,

1 ≤ i ≤ n,

−

where {Si |1 ≤ i ≤ n} is the set of all nonisomorphic simple –modules.

Quivers and Representations

555

7.2. A construction of the elliptic Lie algebra g(X) For the real root system determined by a Cartan datum, one can construct a Lie algebra, namely, the Kac–Moody algebra, that has a nice decomposition of root spaces. The same question can be raised for a generalized root system, especially for an elliptic root system. There have been numerous attempts along these lines, leading to intersection matrix Lie algebras, vertex algebras, toroidal algebras, extended afﬁne Lie algebras, and so on. New advances have been made by K. Saito and D. Yoshii [45], who described simply laced elliptic Lie algebras from three different points of view. The ﬁrst one is from the vertex algebra attached to an elliptic root lattice. In this structure, the root spaces are of an explicit form. The second one is a presentation given by Chevalley generators and generalized Serre relations attached to the elliptic Dynkin diagram. The third one is an amalgamation of an afﬁne Kac–Moody algebra and a Heisenberg algebra. As mentioned in [45], simply laced elliptic Lie algebras are in fact isomorphic to the corresponding 2-toroidal algebras (2-extended afﬁne Lie algebras). More recently, (1,1) (1,1) (1,1) Y. Lin and L. Peng [3] constructed elliptic Lie algebras of type D4 , E6 , E7 , (1,1) or E8 via the root category of the tubular algebras of the same type. In this subsection, we only deal with the following elliptic Dynkin diagrams: 2

1

3

4

6

5

5

4

1 2

5

7

6

8

4

3

1

3

2 8

7

6

9 4

3

1 2

9

8

7

6

5

10

.

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Consider n-dimensional Q-space U, where n is the number of vertices of an elliptic (1,1) (1,1) (1,1) (1,1) Dynkin diagram of type D4 , E6 , E7 , or E8 . Let = {α1 , α2 , . . . , αn } be the canonical basis of U. The symmetric bilinear form (−, −) : U × U → Q attached (1,1) (1,1) (1,1) (1,1) is deﬁned by to an elliptic Dynkin diagram of type D4 , E6 , E7 , or E8 ⎧ 2 if i = j; ⎪ ⎪ ⎨ −1 if there is a real edge between i and j; (αi , αj ) = 2 if there is a double dotted edge between i and j; ⎪ ⎪ ⎩ 0 otherwise. Then the matrix attached to the symmetric bilinear form (−, −), called an elliptic Cartan matrix, is positive semideﬁnite with corank 2. Note that it is not a generalized Cartan matrix in the sense of Kac–Moody algebras [11] because it contains positive entries a1n = an1 = 2 in the off-diagonal part. Deﬁnition 7.2.1 (K. Saito and D. Yoshii). The elliptic Lie algebra of a given elliptic Cartan matrix is deﬁned by Chevalley generators and generalized Serre relations as follows: Generators: {αi | 1 ≤ i ≤ n} and {e±i | 1 ≤ i ≤ n}. Relations: 0. [αi , αj ] = 0, 1 ≤ i, j ≤ n; I. [ei , e−i ] = αi , 1 ≤ i ≤ n; II.1. [αi , ej ] = (αi , αj )ej , for i, j = ±1, ±2, . . . , ±n, where α−i = −αi ; II.2. (ad ei )max{1,1−(αi ,αj )} ej = 0, for i, j = ±1, ±2, . . . , ±n; 1 III. [[ei , e1 ], en ] = 0

for

[[e−i , e−1 ], e−n ] = 0

n

IV. [[[e1 , en ], ei ], ej ] = 0

[[[e1 , e−n ], e−i ], e−j ] = 0

i.

i

[[[e−1 , e−n ], e−i ], e−j ] = 0 [[[e−1 , en ], ei ], ej ] = 0

HH H

1 @

@

for j

@ @

n.

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For a generalized Cartan matrix, the relations of type 0–II are deﬁning relations for a Kac–Moody algebra. But, for an elliptic Cartan matrix, one has to consider additional relations as given by III–IV above. Tubular algebras form a certain special class of associative algebras of global dimension two, which can be determined by quivers with relations. Their module categories and derived categories are of wide interest (see [6,7]), for example, W. Geigle and H. Lenzing [6] found that the derived category of a tubular algebra is isomorphic to the derived category of coherent sheaves on a weighted projective line. (An algebra of the form B = End A (TA ), where A is connected, hereditary, representation–inﬁnite, and TA is a preprojective tilting module, is said to be a concealed algebra. In case A is, in addition, tame, then B is said to be a tame concealed algebra). By deﬁnition, a tubular algebra is a tubular extension of a tame concealed algebra of extension type T(r1 , r2 , . . . , rt ), where (r1 , r2 , . . . , rt ) is equal to (2, 2, 2, 2), (3, 3, 3), (4, 4, 2), or (6, 3, 2). We refer to [7] for a very technical deﬁnition of tubular extension. For a tubular algebra of type T(r1 , r2 , . . . , rt ), its derived category has a nice structure and consists of inﬁnitely many tubular families, where each tubular family is determined essentially by a tame concealed algebra and contains inﬁnitely many homogenous tubes and t nonhomogenous tubes of rank r1 , r2 , . . . , rt , respectively. Y. Lin and L. Peng considered the following tubular algebras given by quivers with relations, that is, = kQ/I:

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For each of the four quivers above, if we forget the orientation and add a dotted edge between the vertices 1 and n (here, n = 6, 8, 9, 10), then the diagram obtained coincides with an elliptic Dynkin diagram. We extend the usual terminology to such (1,1) (1,1) (1,1) (1,1) tubular algebras, calling them of type D4 , E6 , E7 , and E8 . Using the explicit structure of the derived categories of tubular algebras given by D. Happel and C.M. Ringel [46], Y. Lin and L. Peng proved the following theorem: (1,1)

with the isomorphism Theorem 7.2.2 [3]. Let be the tubular algebra of type Xl (1,1) (1,1) (1,1) (1,1) classes of simple modules Si , 1 ≤ i ≤ n, where Xl = D4 , E6 , E7 , or E8 . (1,1) Let g be the elliptic Lie algebra of type Xl , with generators αi and e±i , 1 ≤ i ≤ n, and L(R) the Ringel–Hall Lie algebra over Q of the root category R of . Then, there is a Lie algebra isomorphism : L(R) −→ g given by hSi

→

αi ,

1 ≤ i ≤ n,

uSi

→

ei ,

1 ≤ i ≤ n,

uT (Si )

→

−e−i ,

1 ≤ i ≤ n.

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8. Guide to the literature A wealth of material concerning the representation theory of ﬁnite-dimensional algebras (or quivers) can be found in the books [7, 16, 47] and online notes [14]. For the unexplained notation concerning the theory of Lie algebra, we refer to the book [11], for those concerning quantum groups to the book [28], and for derived categories to the book [42] and the paper [43]. The material related to Ringel–Hall algebras is discussed in the articles [7, 18–21, 27, 39, 40]. For Section 4, we recommend [1,13]. Finally, the articles [2] and [3] contain a detailed exposition of the topics discussed in Section 7. Acknowledgements X. Chen and T. Pospíchal would like to thank Professor Vlastimil Dlab for his help and support. X. Chen is grateful to Professor Jie Xiao for directing him to do research in the ﬁeld of Ringel–Hall algebras. We thank Professor H. Nakajima for sending us his paper [48], which helped us to organize the material in Section 6. References [1] V. Dlab, C.M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), vol. 173, Providence, RI. [2] L. Peng, J. Xiao, Triangulated categories and Kac–Moody algebras, Invent. Math. 140 (2000), 563–603. [3] Y. Lin, L. Peng, Elliptic Lie algebras and tubular algebras, Adv. Math. 196 (2005), 487–530. [4] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447–498. [5] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365–421. [6] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of ﬁnite dimensional algebras, in: G.-M. Greuel, G. Trautmann, eds., Singularities, Representations of Algebras, and Vector Bundles (Lambrecht, 1985), Lecture Notes in Math., vol. 1273, 265–297, Springer–Verlag, Berlin, 1987. [7] C.M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Math., vol. 1099, Springer–Verlag, Berlin–New York, 1984. [8] V.I. Arnold, The A-D-E classiﬁcations, in: Mathematical Developments Arising from Hilbert Problems, Proc. Sympos. Pure Math., vol. 28, American Mathematical Society, Providence, RI, 1976, pp. 46. [9] M. Hazewinkel, W. Hesselink, D. Siersma, F. Veldkamp, The ubiquity of Coxeter-Dynkin diagrams (an introduction to the A-D-E problem), Nieuw Arch. Wisk. 25 (3) (1977), 257–307. [10] J.A. Green, Hall algebras and quantum groups, Textos de Matemática. Série B [Texts in Mathematics. Series B], 4. Universidade de Coimbra, Departamento de Matemática, Coimbra, 1994, pp. vi+23. [11] V. Kac, Inﬁnite dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, UK, 1990. [12] C.M. Ringel, Representation theory of ﬁnite-dimensional algebras, in: Representations of Algebras (Durham, 1985), 7–79, London Mathematical Society Lecture Note Series, vol. 116, Edited by P. J. Webb. Cambridge University Press, Cambridge, UK, 1986.

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[13] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math. 6 (1972), 71–103. [14] W. Crawley-Boevey, Lectures on representations of quivers, lectures given in 1992 at Oxford Univ., available at http://www.amsta.leeds.ac.uk/∼pmtwc/quivlecs.pdf [15] Yu.A. Drozd, Tame and wild matrix problems, (Russian) Representations and quadratic forms (Russian), Akad. Nauk Ukrain. SSR Inst. Mat. Kiev 154 (1979), 39–74. [16] M. Auslander, I. Reiten, S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Math., vol. 36, Cambridge University Press, Cambridge, 1995. [17] V. Kac, Inﬁnite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57–92. [18] C.M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), 583–591. [19] C.M. Ringel, Hall algebras revisited, in: Quantum Deformations of Algebras and their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), 171–176, Israel Math. Conf. Proc., 7, Bar-Ilan Univ., Ramat Gan, 1993. [20] J.A. Green, Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361–377. [21] C.M. Ringel, Green’s theorem on Hall algebras, in Representation Theory of Algebras and Related Topics; CMS Conf. Proc., vol. 19, (1996), 185–245, Amer. Math. Soc., Providence, RI. [22] J. Xiao, Drinfeld double and Ringel–Green theory of Hall algebras, J. Algebra 190 (1997), 100–144. [23] J. Beck, V. Chari, A. Pressley, An algebraic characterization of the afﬁne canonical Basis, Duke. Math. J. 99 (1999), 455–487. [24] J. Beck, H. Nakajima, Crystal bases and two-sided cells of quantum afﬁne algebras, Duke Math. J. 123 (2004), 335–402. [25] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), 465–516. [26] Z. Lin, J. Xiao, G. Zhang, Representations of tame quivers and afﬁne canonical bases, preprint, arXiv:0706.1444. [27] C.M. Ringel, PBW–bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. [28] G. Lusztig, Introduction to quantum groups, Progress in Math., vol. 110, Birkhäuser, Boston, 1993. [29] V. Chari, A. Pressley, Quantum afﬁne algebras at roots of unity, Representation Theory 1 (1997), 280–328. [30] I. Damiani, A basis of type Poincaré–Birkhoff–Witt for the quantum algebra , sl(2), J. Algebra 161 (1993), 291–310. [31] F. Gavarini, A PBW basis for Lusztig’s form of untwisted afﬁne quantum groups, Comm. Algebra 27 (1999), 903–918. [32] I.N. Bernstein, I.M. Gelfand, V.A. Ponomarev, Coxeter functors and Gabriels theorem, Uspekhi Math. Nauk 28 (1973), 19–33. [33] X. Chen, J. Xiao, Root vectors arising from Auslander-Reiten Quivers, J. Algebra 222 (1999), 328–356. [34] X. Chen, J. Xiao, Exceptional sequence in Hall algebra and quantum group, Compositio Math. 117 (1999), 161–187. [35] X. Chen, Root vectors of the composition algebra of the Kronecker algebra, Algebra Discrete Math. 1 (2004), 37–56. [36] K. McGerty, The Kronecker quiver and bases of quantum afﬁne sl2 , Adv. Math. 197 (2005), 411–429. [37] P. Zhang, PBW–bases of the composition algebra of the Kronecker algebra, J. Reine Angew. Math. 527 (2000), 97–116. [38] P. Baumann, C. Kassel, The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207–233. [39] C.M. Ringel, Hall algebras, in Topics in Algebra, Part 1 (Warsaw, 1988), 433–447, Banach Center Publ., 26, Part 1, PWN, Warsaw, 1990. [40] C.M. Ringel, Hall polynomials for the representation-ﬁnite hereditary algebras, Adv. Math. 84 (1990), 137–178.

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[41] D. Happel, On the derived category of a ﬁnite-dimensional algebra, Comment. Math. Helv. 62 (1987), 339–389. [42] D. Happel, Triangulated categories in the representation theory of ﬁnite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, UK, 1988. [43] B. Keller, Derived categories and their uses, Handbook of algebra, vol. 1, 671–701, North-Holland, Amsterdam, 1996. [44] C.M. Ringel, The braid group action on the set of exceptional sequences of a hereditary Artin algebra, in: Abelian Group Theory and Related Topics (Oberwolfach, 1993), 339–352, Contemp. Math., 171, Amer. Math. Soc., Providence, RI, 1994. [45] K. Saito, D. Yoshii, Extended afﬁne root systems IV (Simple laced elliptic Lie algebras), Publ. RIMS. Kyoto Univ. 36 (2000), 385–421. [46] D. Happel, C.M. Ringel, The derived category of a tubular algebra, Lecture Notes in Math., vol. 1273, 156–180, Springer–Verlag, Berlin–Heidelberg–New York, 1986. [47] I. Assem, D. Simson, A. Skowro´nski, Elements of the representation theory of associative algebras, vol. 1: Techniques of representation theory. London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge, 2006. [48] H. Nakajima, Crystal, Canonical and PBW bases of quantum afﬁne algebras, Algebraic groups and homogeneous spaces, 389–421, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007.

Canonical Decompositions and Invariants for Data Analysis Marlos Viana University of Illinois at Chicago Eye Center, Chicago, IL 60612, USA E-mail: [email protected]

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A general principle of interpretability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetry studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The regular invariants of S3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The canonical projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Interpreting the regular invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. A set of bases for the regular invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. A numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The regular invariants of S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. The regular invariants in dimension of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The regular invariants in dimension of 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Comments and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

HANDBOOK OF ALGEBRA, VOL. 6 Edited by M. Hazewinkel Copyright © 2009 by Elsevier B.V. All rights reserved DOI: 10.1016/S1570-7954(08)00210-6 565

566 566 567 571 572 573 574 576 576 577 580 582 584

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1. Introduction Riley et al. [3] observed that If a physical system is such that after application of a particular symmetry transformation the ﬁnal system is indistinguishable from the original system, then its behavior, and hence the functions that describe its behavior, must have the corresponding property of invariance when subject to the same transformations. The study of these transformations is a study of the symmetries of the system. More generally, as Bacry [1] shows, the study of the symmetries of a physical system often suggests the study of the symmetries of certain physical laws and theories, and frequently, leads to symmetry-related principles in the form of conservation laws. In this chapter, a data analytic interpretation of these principles is proposed. To illustrate, consider the group G = {1, v, h, o} of vertical (v) and horizontal (h) line reﬂections, point reﬂection (o), and the identity (1) under composition of functions. Its multiplication table ∗ 1 v h o

1 1 v h o

v h v h 1 o o 1 h v

o o h v 1

(1.1)

shows that G is Abelian. Of course, these are the automorphisms of a rectangle. Examples of data indexed by these transformations appear, for example, in [5] and [6, Chapter 1]. Indicating by {x1 , xv , xh , xo } the data indexed by these transformations, it then follows that the four scalar quantities, I1 = x1 + xo + xv + xh , Iv = x1 + xv − xo − xh , Ih = x1 + xh − xo − xv , and Io = x1 + xo − xv − xh ,

(1.2)

are the invariants of interest in the data space under the regular group action. These data summaries vary only up to companion points in the same invariant subspace. For example, the invariant x1 + xo − xv − xh is such that xt∗1 + xt∗o − xt∗v − xt∗h = ±(x1 + xo − xv − xh ) for all

t ∈ G.

Ideally, then, it is of interest to interpret and describe both the underlying group actions and the resulting invariants. 2. A general principle of interpretability As a general principle, group actions can be interpreted as degrees of arbitrariness, or irrelevance, that are chosen upon the labels and need to be resolved. For example,

Canonical Decompositions and Invariants for Data Analysis

567

it can be shown that the sum and n − 1 sets of pairwise comparisons in a sample of size n are the invariants that result when the assignment of observations to the labels {1, 2, . . . , n} is (determined to be) arbitrary in such a way that its degree of arbitrariness amounts to shufﬂing the labels in all possible n! ways. These invariants, in dimensions of 1 and n − 1, respectively, appear in the practice of data analysis as the sample mean and its standard deviation. If, however, the data are labeled by n equally spaced points along a circle and the degree of arbitrariness is now restricted to shufﬂing the labels according to the n cyclic permutations, then the resulting invariants are the n linear combinations of the original data, with as coefﬁcients the n sets of Fourier coefﬁcients cos(2πjk/n) + i sin(2πjk/n), j, k = 0, . . . , n − 1, that is, the n ﬁnite Fourier transforms of the observed data. Applying this principle to the example introduced in Section 1, the invariants (1.2) can be characterized as those data that should be retained when the arbitrariness of where is left (right) and where is up (down) is resolved. This degree of arbitrariness can be identiﬁed by studying the group action, deﬁned by the multiplication table in (1.1), each line of which relabels what is up and down and what is left–right. It can be referred to as planar orientation so that then the pairwise comparisons in (1.2) are the invariants of planar orientation. For example, x1 + xo − xv − xh gives a onedimensional subspace regardless of the planar orientation. Equivalently, it compares the responses to point symmetry against the combined responses of vertical and horizontal responses, regardless of the planar orientation. In the present case, all four invariants are in dimension of one, adding to the dimension (4) of the original vector space for the data indexed by the elements of G. A simpler example is given by the invariants x ± xr associated with bilateral, say, left () and right (r) measurements, determined by the identity transformation and the transposition of left and right measurements. Here, left–right is the arbitrariness resolved by x ± xr . When an invariant is derived from a regular group action (the group acting on itself by multiplication on the right or on the left), it is called a regular invariant. In the following sections, the data analytic methods of symmetry studies will be applied to determine and interpret the regular invariants associated with experimental data indexed by the permutations of three and four objects. 3. Symmetry studies Methods of representation theory can be systematically applied in the analysis of data indexed by a set of labels where a group action can be deﬁned and often justiﬁed by an experimental context. While group representations provide the connection between the labels and the data space, the canonical decomposition theorem, reducing the identity matrix in the data space to a sum of algebraically orthogonal projections, provides the conditions with which the statistical theory of quadratic forms can be formulated to study the resulting decomposition of the sum of squares or analysis of variance. The study of these projections also leads to useful summaries of the data in

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the form of canonical invariants, which can be used as descriptive tools or interpreted for further statistical inference. These basic elements constitute the core of all symmetry studies as introduced by the author in [6], where a detailed introduction to a number of data analytical applications is described. Speciﬁcally, every symmetry study includes the identiﬁcation of the following: (1) A set V of labels with v elements; (2) The observed data x deﬁned in a linear subspace V ⊂ Rv and indexed by those labels (the structured data); (3) A rule or group action, according to which the symmetry transformations in a group G are applied to V ; (4) The classes and multiplicities of the resulting elementary orbits, subsets of V , on which G acts transitively; (5) The resulting linear representations of these actions in the corresponding data vector subspaces; (6) The canonical projection matrices Pχ indexed by the irreducible characters χ of G; (7) The canonical invariants Pχ x on the data x, and their interpretations; (8) A statistical analysis of the invariants and, if applicable, their canonical x = P x based on the canonical decomposition analysis of variance x x χ χ I = χ Pχ of the identity operator in the data subspaces, as a sum of algebraically orthogonal projections. When necessary, these canonical projections may be grouped together to describe V. Speciﬁcally, if there are qλ orbits of type λ, each with oλ elements, classifying the v = λ qλ oλ points in V, then for each class of orbits with vλ = qλ oλ points, the identity matrix in V reduces according to Ivλ = Pλ1 +Pλ2 +. . ., where Pλj is the canonical projection constructed with qλ copies of the canonical projection Pjλ in the vector space associated with the elementary orbits of type λ and indexed by the irreducible representations j = 1, 2, . . . of the underlying group G. The reduction in the original space V is then Iv = Diag(. . . , Ivλ , . . .), with as many components as the different types of orbits identiﬁed by the action of G on V . To illustrate, consider the set V = {uu, yy, uy, yu} of binary sequences in length of two or, equivalently, the set of all mappings s from {1, 2} into {u, y} so that ϕ1 (τ, s) = sτ −1 , τ −1

s

→ {u, y}, s ∈ V, τ ∈ S2 = {1, (12)}, ϕ1 : {1, 2} −−→ {1, 2} −

(3.1)

and ϕ2 (σ, s) = σs, s

σ

→ {u, y} − → {u, y}, s ∈ V, σ ∈ S2 = {1, (uy)}, ϕ2 : {1, 2} −

(3.2)

are actions of S2 on V . The action ϕ1 classiﬁes the sequences up to permutations of the positions {1, 2} of the letters, whereas ϕ2 classiﬁes the sequences up to permutations of the symbols {u, y}. The action ϕ1 generates three orbits {uu}, {yy}, and {uy, yu} and ϕ2 generates two orbits {uu, yy} and {uy, yu}.

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569

In the position symmetry case, the representation is isomorphic to 1 0 0 1 ξ1 = Diag 1, 1, , ξt = Diag 1, 1, Diag 1, 1, , t ≡ (12), 0 1 1 0 taking values in GL( R4 ). Equivalently, writing V = R4 , it is observed that the ξ determines the decomposition V = W1 ⊕ W2 ⊕ W3 , where the stable subspaces W1 and W2 reduce isomorphically as a subspace U1 associated with the unit representation, and W3 reduces as the sum of an isomorphic copy of U1 and a subspace USgn associated with the sign (Sgn) representation. Therefore, V = U1 ⊕ U1 ⊕ U1 ⊕ USgn = V1 ⊕ VSgn ,

(3.3)

V1

showing a decomposition of V into a direct sum of the irreducible representations of S2 in which the isomorphic copies are collected together. This is the canonical decomposition of V. Next, to construct the projections P1 and PSgn of V on the irreducible subspaces V1 and VSgn , deﬁne Pβ = nβ χβ (τ)ξτ /g, τ∈G

where χβ is the irreducible character of the irreducible representation β of G with |G| = g elements and nβ its dimension. In the present case (G = S2 ), β ∈ {1, Sgn} with corresponding characters χ1 ≡ 1 and χSgn = Sgnτ so that P1 = (ξ1 + ξt )/2 and PSgn = (ξ1 − ξt )/2. When the projections are evaluated relative to a basis of V = V1 ⊕ VSgn on which ξτ = Diag(I3 ⊗ 1τ , Sgnτ ), where In indicates the n × n identity matrix, the results are Diag(1, 1, 1, 0) and PSgn = Diag(0, 0, 0, 1). 2 = P It then follows that (1) P12 = P1 , PSgn Sgn ; (2) P1 PSgn = PSgn P1 = 0, and (3) I4 = P1 + PSgn so that Pβ is a projection on a subspace isomorphic to Vβ . These properties remain valid if Pβ is evaluated relative to any representation Mξτ M −1 equivalent to ξ. In this case, Pβ transforms as MPβ M −1 and (1), (2), and (3) remain unchanged. For example, relative to the basis for V indexed by {uu, yy, uy, yu}, P1 = Diag(1, 1, A2 );

PSgn = Diag(0, 0, Q2 ),

where An indicates the n × n matrix with all entries equal to 1/n and Qn = In − An . Similarly, in the letter-symmetry case, ξ is given by 0 1 0 1 ξ1 = Diag(I2 , I2 ), ξt = Diag , , 1 0 1 0 so that now V = W1 ⊕ W2 and in each one of these two stable subspaces ξ reduces isomorphically as the sum of the unit and the sign representations. Collecting the

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isomorphic copies of the corresponding irreducible subspaces U1 and USgn , it follows that V = U1 ⊕ U1 ⊕ USgn ⊕ USgn ,

V1

VSgn

which is the canonical decomposition of V, with corresponding projections P1 = (ξ1 + ξt )/2 and PSgn = (ξ1 − ξt )/2. When these matrices are evaluated relative to a basis of V = V1 ⊕ VSgn on which ξτ = Diag(I2 ⊗ 1τ , I2 ⊗ Sgnτ ), the result is P1 = Diag(1, 1, 0, 0) and PSgn = Diag(0, 0, 1, 1). It then follows that Pβ is a projection on a subspace isomorphic to Vβ and, regardless 2 =P of the chosen basis for V, P12 = P1 , PSgn Sgn ; P1 PSgn = PSgn P1 = 0 and I4 = P1 + PSgn . In particular, P1 = Diag(A2 , A2 ),

PSgn = Diag(Q2 , Q2 ).

The arguments illustrated in the above example lead to a proof of following result [4, 6]: Theorem 3.1 (Canonical decomposition). Let ρ be a linear representation of G into GL(V), and let ρ1 , . . ., ρh be the distinct nonisomorphic irreducible representations of G with g elements, with corresponding characters χ1 , . . ., χh and dimensions n1 , . . ., nh . Then, Pi = ni χi (τ)ρτ /g τ∈G

is a projection of V onto a subspace Vi , sum of mi isomorphic copies of the irreducible subspaces associated with ρi , i = 1, . . ., h. Moreover, Pi Pj = 0, for i = j, Pi2 = Pi and i Pi = Iv , where v = dim V = hi=1 mi ni . The Fourier inverse formula In several experiments, however, the data are naturally indexed by G in itself so that V = G. In this case, there is a one-to-one correspondence x(β) = xτ βτ ←→ xτ = nβ tr [βτ −1 x(β)] τ∈G

β∈G

x(β) over the between the experimental data {xτ , τ ∈ G} and the Fourier transforms irreducible representations β of G, in dimension of nβ , where in the above expression, indicates the set of all irreducible representations of G. Observe, in particular, that G the (regular) invariants in (1.2) are the Fourier transforms of the data indexed by G. More generally, it is given as follows. Proposition 3.1. The rows of the Fourier transform x(β) matrix reduce as β under the regular action.

Canonical Decompositions and Invariants for Data Analysis

571

Proof. Assume, without loss of generality, that β is in dimension of 2; select the ﬁrst row of x(β) and write βijτ to indicate the ij entry of βτ . The objective here is to show that G acting on the components τ τ τβ11 , v= τβ12 u= τ∈G

τ∈G

of the ﬁrst row of x(β) as points in the group algebra of G according to the regular action (uσ −1 , vσ −1 ) gives βσ so that the representation coincides with (one of the two copies of ) β, that is, σ σ + vβ21 , uσ −1 = uβ11

Indeed, σ σ uβ11 + vβ21 =

σ σ vσ −1 = uβ12 + vβ22 .

τσ γ −1 τ σ τ σ τ β11 β11 + β12 β21 = τβ11 = (γβ11 )σ = uσ −1 τ

τ

γ

σ + vβσ = vσ −1 , proving the proposed equality and concluding the and similarly uβ12 22 proof.

Replacing τ by xτ (the scalar data indexed by G) so that the rows of the Fourier is transform x(β) should give a basis for an invariant subspace tagged by β. If β ∈ G in dimension of m, then there are m such bases, corresponding to the m copies with which β appears in the regular representation of G. Repeating the same argument for the remaining rows of x(β), it is seen that, in matrix form, Proposition 3.1 can be expressed as xτσ βτ . (3.4) x(β)βσ −1 = τ

In particular, the rows (or columns if the regular action is from the right) form a set of bases for two isomorphic copies of β. In the next two sections, these results will be applied to the study of the regular invariants of S3 and S4 . 4. The regular invariants of S3 Consider the observed frequencies, shown in Table (4.1), with which the DNA words in the permutation orbit Oact = {act, tac, cta, cat, tca, atc} = {sτ −1 ; τ ∈ S3 } of the mapping s = act appear in nine subsequent 900-bp-long regions along the BRU isolate of the human immunodeﬁciency virus type I. The entire 9229-bp-long single-strain DNA sequence is available from the National Center for Biotechnology Information1 data base using the accession number K02013. 1 http://www.ncbi.nlm.nih.gov/.

572

M. Viana

It is assumed in this section that the elements of S3 are written in the order of {1, (123), (132), (12), (13), (23)} so that Oact is obtained by evaluating sτ −1 in that order, where s(1) = a, s(2) = c, and s(3) = t. In what follows, moreover, the components of vectors such as x = (act, tac, cta, cat, tca, atc) indicate in short notation the data and at the same time the corresponding permutations in S3 . Other common examples occur in ranking preferences studies (e.g. [2]) or scalar data in psychometric studies, where scalar responses to the different sequences of presentation of a stimuli are recorded. 1 act 8 tac 7 cta 15 cat 14 tca 11 atc 7 total 62

2 16 17 8 15 18 15 89

3 16 13 14 16 10 9 78

4 5 6 7 17 11 15 9 11 9 14 15 14 21 17 17 11 16 13 11 11 75 83 81

7 8 12 6 18 5 8 5 15 10 14 9 11 12 78 47

9 14 17 16 8 13 10 78

(4.1)

Having identiﬁed the labels and the structured data, it is remarked here again that their notation will often be identiﬁed so that, for example, act ≡ xact and act + tac is to be interpreted as xact + xtac . In particular, the case in which the permutations in S3 act on the labels (which are in one-to-one correspondence with S3 ) will be studied. The regular action will be deﬁned by its multiplication table 1 2 3 4 5 6

1 1 2 3 4 5 6

2 2 3 1 5 6 4

3 3 1 2 6 4 5

4 4 6 5 1 3 2

5 5 4 6 2 1 3

6 6 5 4 , 3 2 1

(4.2)

where the permutations 1, (123), (132), (12), (13), and (23) are identiﬁed with 1, 2, 3, 4, 5, and 6, respectively. The resulting representation is the regular representation of S3 , indicated here by φ. For example, when τ = (12), φτ is the permutation matrix deﬁned by rows 1 and 4 of Table (4.2). Because this action is transitive, Step (4) in the summary presented in Section 3 leads to a simple single-orbit symmetry study. 4.1. The canonical projections Indicate the irreducible representations of S3 by 1, the symmetric or trivial representation; by Sgn, the alternating (or sign) representation (both in dimension of 1); and

Canonical Decompositions and Invariants for Data Analysis

573

by β in dimension of two. Their table of characters is given by τ χ1 1 1 (123) 1 (132) 1 (12) 1 (13) 1 (23) 1

χSgn 1 1 1 −1 −1 −1

χβ 2 −1 −1 , 0 0 0

from which the evaluation of the canonical projections Pχ = nχ χ(τ)φτ /6 τ∈G

gives P1 the 6 × 6 matrix with entries equal to 1/6, ⎡ ⎤ 1 1 1 −1 −1 −1 ⎢ 1 1 1 −1 −1 −1 ⎥ ⎢ ⎥ ⎢ 1 1 1 −1 −1 −1 ⎥ ⎢ ⎥ PSgn = 1/6 ⎢ ⎥, 1 1 1⎥ ⎢ −1 −1 −1 ⎢ ⎥ ⎣ −1 −1 −1 1 1 1⎦ −1 −1 −1 1 1 1 ⎡

2 ⎢ −1 ⎢ ⎢ −1 Pβ = 1/3 ⎢ ⎢ 0 ⎢ ⎣ 0 0

−1 2 −1 0 0 0

⎤ −1 0 0 0 −1 0 0 0⎥ ⎥ 2 0 0 0⎥ ⎥. 0 2 −1 −1 ⎥ ⎥ 0 −1 2 −1 ⎦ 0 −1 −1 2

4.2. Interpreting the regular invariants Borrowing language from physics [1], the permutations {1, (123), (132)} will be referred to as rotations and {(12), (13), (23)} as reversals. The terminology is a consequence of the action of S3 on the ordered edges of a regular triangle, which identiﬁes the three-fold rotations seen from each side of the triangle’s plane. Applying the interpretability principles introduced in Section 2, it is observed that (1) P1 x gives the symmetric invariant act + tac + cta + cat + tca + atc; (2) PSgn x gives the alternating invariant act + tac + cta − cat − tca − atc, comparing the total effect of rotations with the total effect of reversals, and (3) Pβ x evaluates as an invariant of within-rotation variability and within-reversal variability, giving an invariant subspace in dimension of four.

574

M. Viana

The degree of arbitrariness induced by the regular action can be interpreted as the arbitrariness intrinsic to the notion of rotations and reversals, which in a sense is equivalent to a notion of planar orientation. To see this, observe that although there are 3! = 6 distinct assignments of the symbols {a, c, t} to the labeled (as 1, 2, 3, say) vertices of a regular triangle, there are only two distinct assignments to those vertices when their labels are removed (by shufﬂing): namely to the two distinct classes {(a, c), (c, t), (t, a)} and {(c, a), (a, t), (t, c)} of oriented edges. These two classes are the rotations and the reversals. It is simple to verify that the action of the permutations act, tac, cta, cat, tca, and atc on these edges replicates precisely the regular action of S3 . With this language, it can be said that (1)–(3) above are the canonical invariants of rotations and reversals. It is remarked however that, more generally, (1)–(3) are the regular invariants of S3 . 4.3. A set of bases for the regular invariants It is simple to verify that 1 0 0 1 −1 −1 0 β: , , , 0 1 −1 −1 1 0 1 1 0 , −1 −1

1 −1 , 0 0

−1 , 1

where the matrices follow the order of S3 described above,gives an irreducible repx(β) = τ xτ βτ of the structured resentation of S3 . The resulting Fourier transform data x at β is then act − cta − tca + atc tac − cta + cat − tca x(β) = β ⊕ β. −tac + cta + cat − atc act − tac + tca − atc By Proposition 3.1, it identiﬁes the two bases Iβ = {act − cta − tca + atc, tac − cta + cat − tca} and Iβ = {−tac + cta + cat − atc, act − tac + tca − atc} from rows 1 and 2 of x(β), respectively. The two (isomorphic) subspaces spanned by these two bases reduce as β under the regular action of S3 , that is, (3.4) is obtained. The alternating invariant Clearly, a trivial application of Proposition 3.1 with the alternating character gives the invariant ISgn = act + tac + cta − cat − tca − atc comparing rotations and reversals.

Canonical Decompositions and Invariants for Data Analysis

575

The vectors {I1 , ISgn , Iβ , Iβ } deﬁne the (rows of the) change of basis matrix ⎡ ⎤ 1 1 1 1 1 1 ⎢ 1 1 1 −1 −1 −1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1 −1 0 0 −1 1 ⎥ ⎢ ⎥, B=⎢ 1 −1 1 −1 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎣ 0 −1 1 1 0 −1 ⎦ 1

−1

0

0

1 −1

relative to which BP1 B−1 = Diag(1, 0, 0, 0, 0, 0), BPSgn B−1 = Diag(0, 1, 0, 0, 0, 0), and BPβ B−1 = Diag(0, 0, 1, 1, 1, 1). Similarly, the regular action of S3 extended to the components of {I1 , ISgn , Iβ , Iβ } gives the representation φ˜ τ = Diag(1, Sgnτ , I2 ⊗ βτ ), relative to which the canonical projections are, respectively, BP1 B−1 , BPSgn B−1 , and BPβ B−1 . Interpreting voting preferences In the context of voting preferences, in which the data are the frequencies with which voters select among all permutations of the rankings of three candidates or objects, it is observed that Iβ = {act − cta − tca + atc, tac − cta + cat − tca} ≡ {v1 , v2 } can be characterized (as highlighted by a) as a basis for the space jointly describing “more radical” (v1 , ﬁrst and third rank) and “less radical” (v2 , second and third rank) comparisons. The same interpretation, not surprisingly, applies to Iβ , with the symbols c and a interchanged. From Proposition 3.1, these two position-comparisons depend on the symbols only up to linear superpositions of the components of Iβ . For example, the difference of these two components compares ﬁrst and second ranks. Interpreting planar rotations and reversals Following from Section 4.3, the regular invariants ISgn and Iβ (or Iβ ) can be interpreted as (1) the comparison between rotations and reversals, in dimension 1; (2) the two-component within-rotation, within-reversal variability, in dimension 2 (each). More speciﬁcally, indicating by → and ← the two opposing (chiral) orientations of planar rotation, it is observed that v1 = act − cta + atc − tca , and v2 = tac − cta + cat − tca . ← two-fold

→ one-fold

← one-fold

→ two-fold

576

M. Viana 1.0

Variability

0.5

0.0

⫺0.5 ⫺1.0

V_2 V_1 1

3

5

7

9

Region

Fig. 1. The components of the regular invariant Iβ (chiral variability) along the BRU isolate.

Each component is a within-rotation, within-reversal comparison. However, they are distinguished by the way with which direction and phase interact. They describe a right-hand two-fold followed by a left-hand one-fold screw motion, and a lefthand one-fold followed by a right-hand two-fold screw. It is suggested here that this property be interpreted as chiral variation.

4.4. A numerical example Figure 1 displays the two components v1 = act − cta − tca + atc and v2 = tac − cta + cat − tca of the regular invariant Iβ for the DNA (log-transformed) frequency data from Table (4.1) along the nine subsequent regions of the genome, suggesting the presence of a dominant direction of higher (chiral) variability.

5. The regular invariants of S4 The complete symmetry study of data indexed by S4 is the study of ﬁve vector (invariant) subspaces, V = V1 ⊕ VSgn ⊕ 2Vη ⊕ 3Vβ ⊕ 3Vγ , reducing1 the original data space (see also [2]). Each of the three copies of the subspaces Vβ and Vγ are of dimension three, whereas the two copies of Vη are of dimension two. There are, in addition, the alternating (VSgn ) and the trivial (V1 ) subspaces, 1 If G has g elements and its r irreducible representations occur with multiplicities d then g = d 2 + i 1 d22 + · · · + dr2 . E.g. [5].

Canonical Decompositions and Invariants for Data Analysis

577

both of dimension 1. In what follows, a set of bases {Iβ , Iβ , Iβ }, {Iγ , Iγ , Iγ }, and {Iη , Iη } for 3Vβ , 3Vγ , and 2Vη , respectively, will be described. It is assumed, from now on, that the elements of S4 are listed sequentially following the rows of Matrix (5.1)

S4 :

1 (12) (34) (123) (143) (124) (1324) (12)(34)

(13) (14) (132) (234) (142) (1234) (1423) (1243)

(23) (243) (13)(24) (14)(23)

(24) (134) (1432) (1342)

(5.1)

so that when S4 acts (via sτ −1 ) on the positions {1, 2, 3, 4} of the symbols in s = agct, the resulting complete set of labels (also indicating the data x indexed by those labels) is given by Matrix (5.2).

x:

agct agtc cgta tcag

gact cagt tacg gatc

cgat gcat gtca ctga

tgca atgc tagc catg

acgt actg ctag tcga

atcg tgac gcta gtac

.

(5.2)

5.1. The regular invariants in dimension of 2 Following Viana [6, Section 3.9], a two-dimensional irreducible representation of S4 , corresponding to the partition λ = 22, can be obtained from the transitive action of S4 on the classes 1|2 , 3|4

1|3 , 2|4

1|4 , 2|3

2|3 , 1|4

2|4 , 1|3

3|4 1|2

of equivalent Young diagrams, shown in Table (5.3), where the classes are labeled as 1, 2, . . . , 6, respectively. The action gives a permutation representation, indicated here by ξ, with which the projections Pi = Sgnτ ξτ τ∈Gi

in dimension of one based on the column stabilizers G1 = {1, (12), (34), (12)(34)}, G2 = {1, (13), (24), (13)(24)}, and G3 = {1, (14), (23), (14)(23)} identify the bases u=

1|3 1|4 2|3 2|4 − − + ≡2−3−4+5 2|4 2|3 1|4 1|3

v=

1|2 1|4 2|3 3|4 − − + ≡ 1 − 3 − 4 + 6. 3|4 2|3 1|4 1|2

578

M. Viana

The action of S4 on {u, v}, shown in the two rightmost columns of Table (5.3), gives the representation (η) of interest. The resulting matrix representation and its table of characters are shown in (5.4) and (5.5), respectively, corresponding to the elements of S4 shown on Table (5.1). 1|2 3|4

1|3 2|4

1|4 2|3

2|3 1|4

2|4 1|3

3|4 1|2

1 (12) (13) (14) (23) (24) (34) (123) (234) (134) (124) (1234) (1324) (1243)

1 1 4 5 2 3 1 4 2 4 5 4 6 5

2 4 2 6 1 2 3 5 3 6 4 5 4 1

3 5 6 3 3 1 2 1 1 2 6 1 2 4

4 2 1 4 4 6 5 6 6 5 1 6 5 3

5 3 5 1 6 5 4 2 4 1 3 2 3 6

6 6 3 2 5 4 6 3 5 3 2 3 1 2

η:

1 0 0 1 −1 0 −1 1 0 −1 1 −1 −1 0 −1 1

τ

−1 0 −1 1 −1 1 −1 0 0 −1 1 −1 1 0 0 1

1 0 0 1 −1 −1 −1 −1

−1 −1 −1 −1 1 0 0 1

0 1 0 1 1 0 0 1

1 0 −1 −1 −1 −1 1 0

0 1 −1 −1 1 0 1 0

u

v

u v −u −u + v u−v −v v u v u u−v −v −u −u + v −u + v −u −v u−v −u + v −u −v u−v u−v −v −u −u + v v u

1 1 0 0 1 −1 0 −1 0 1 1 0 0 0 1 1

−1 −1 1 0 −1 −1 1 0

(5.3)

(5.4)

Bases for the regular invariants in dimension of two Applying Proposition 3.1 to the irreducible representation η, with the labels in (5.1), gives (from the rows of the Fourier transform of x at η) the bases {v1 , v2 } and {w1 , w2 } shown, in matrix form, in (5.6)–(5.9).

χη :

2 0 −1 0

0 0 0 0 0 −1 −1 −1 −1 −1 −1 −1 0 2 0 2 0 0 2 0

,

(5.5)

Canonical Decompositions and Invariants for Data Analysis

⎡

agct ⎢−agtc v1 = ⎢ ⎣ 0 −tcag

⎤ −gact cgat 0 0 atcg −cagt 0 0 −actg −tgac⎥ ⎥, 0 −gtca tagc ctag gcta⎦ gatc −ctga 0 tcga 0

⎡

acgt actg 0 0

⎤ −atcg tgac⎥ ⎥, −gcta⎦ gtac

tgca acgt atgc −actg 0 0 catg 0

⎤ 0 −tgac⎥ ⎥, 0⎦ gtac

0 −atgc −tagc 0

⎤ −atcg 0⎥ ⎥. −gcta⎦ 0

0 0 −cgat tgca ⎢ 0 cagt −gcat −atgc v2 = ⎢ ⎣−cgta −tacg gtca −tagc 0 0 0 catg ⎡

0 ⎢−agtc w1 = ⎢ ⎣ cgta −tcag

579

−gact 0 −cagt gcat tacg −gtca 0 −ctga

⎡

agct gact −cgat ⎢ agtc 0 −gcat w2 = ⎢ ⎣−cgta −tacg 0 tcag gatc ctga

0 0 ctag tcga

(5.6)

(5.7)

(5.8)

(5.9)

Interpretation Write uv = umvn+unvm, where u, v, m, and n are distinct symbols in {a, g, c, t} so that, for example, ac = agct + atcg. With that notation in mind, the following diagram illustrates the 16 components of w1 (Note that uv sums over nonadjacent positions in the square):

a

c

a

t

a

g

a

c ag at

t

g

c

g

c

t

g

t

g

c

g

t

g

a

g

t ga gc

t

a

c

g

c c

a a

t

c

a

c

c

a

c

t

a

t

g

t

t

g

a

g

t

g

t

a

t

g

t

c

a

c

g

c

c

a

g

a

ct

cg

tc

ta

580

M. Viana

Similarly, redeﬁning uv = uvmn + uvnm, summing now over two adjacent positions in the square, it follows that w2 is given by

a

g

a

t

a

g

a

t ag at

t

c

g

c

c

t

c

g

g

a

g

c

g

c

g

a ga gc

t

c

t

a

a

t

c

t

c

g

c

g

c

t

c

t ct cg

t

a

a

t

g

a

a

g

t

a

t

a

t

c

t

c tc

c

g

g

c

a

g

a

ta

g

Similar interpretation is obtained for the basis {v1 , v2 }. 5.2. The regular invariants in dimension of 3 Table (5.10) shows one of the two irreducible representation of S4 of dimension three [6] indexed in correspondence with the elements of S4 in Table (5.1). Applying Proposition 3.1 to β with the labels in (5.1), the resulting bases (rows of the Fourier transform of x at β) are shown in (5.11), (5.12), and (5.13), indicated respectively by Iβ , Iβ , and Iβ , β= 1 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 −1 −1 0 0 0 1 0 0 −1 1 0 0 −1 −1 0 −1

−1 0 0 0 −1 0 0 1 1 0 −1 0 0 0 1 0 1 0 0 1 −1 0 1 0 −1 −1 0 1 −1 −1 0 1 0 0 0 0 −1 1 1 0 1 0 0 0 1 −1 0 −1 0 0 1 −1 1 1 0 0 −1 0 −1 1 0 0 −1 −1 0 0 0 1 −1 0 1 0 1 −1 −1 0 −1 1 1 0 0 −1 0 0

0 −1 1 1 1 0 0 −1 0 0 0 −1 1 0 1 0 −1 1 0 0 −1 0 0 1 1 0 0 −1 −1 1 −1 0 −1 0 0 0 0 1 0 1 0 −1 0 −1 1 −1 −1 −1

0 1 0 1 0 0 1 −1 1 0 0 −1 1 0 −1 0 −1 0 0 1 −1 −1 0 0 . 1 −1 0 −1 −1 1 0 0 −1 1 1 0 −1 −1 −1 0 0 1 0 1 0 1 0 0

(5.10)

Canonical Decompositions and Invariants for Data Analysis

581

⎡

⎤ agct − gact + acgt + atcg + agtc − gcat + actg + atgc − gtca − gcta − gatc − gtac ⎢ ⎥ ⎣−cgat + acgt − gcat + cagt + atgc − tgac − ctag + tagc + ctga − tcag − gtac + tcga⎦, (5.11) −tgca + atcg + actg − cgta − gtca + tacg − gcta + ctag − ctga + tcag − tcga + catg ⎡ ⎤ gact − cgat + gcat − cagt − cgta + gtca + gcta − ctag − ctga + gatc + gtac − catg ⎢ ⎥ ⎣ agct + gact + tgca − acgt + atcg − cagt − atgc + gtca + tacg − tagc − ctga − tcga⎦, (5.12) −atcg + agtc − actg + atgc + tgac − tacg − ctag + tagc + gatc − tcag + gtac − catg ⎡ ⎤ gact − tgca + gcat − tgac + gtca − tacg + gcta − tagc + gatc − tcag + gtac − tcga ⎢ ⎥ ⎣−acgt + agtc − cagt + actg − atgc + cgta + gcta − tagc − ctga + gatc − tcga + catg⎦. (5.13) agct + gact + cgat + acgt − atcg + gcat + cagt − actg − tacg − ctag − tcag − catg

each one spanning one copy of Vβ . Inspection of these bases shows that (1) Iβ describes “g versus a” at positions 1, 3, 4; (2) Iβ describes “g versus c” at positions 1, 3, 4; (3) Iβ describes “g versus t” at positions 1, 3, 4. The position 2, not present above, is in fact g’s position in the generating sequence “agct.” In each subspace, the three coordinates correspond to positions 1, 3, 4. The reduction in the second invariant space in dimension three The companion (distinct) irreducible representation of S4 is simply γτ = Sgnτ βτ . It is obtained by multiplying the matrices βτ in (5.10) by the parity Sgnτ of the corresponding permutation. Consequently, the bases for the invariant subspaces of interest are the rows of Fγ = τ xτ Sgnτ βτ , ⎡

⎤ agct + gact − acgt − atcg − agtc − gcat + atgc + actg − gtca + gcta − gatc + gtac ⎢ ⎥ ⎣cgat − acgt + cagt − gcat + atgc − tgac − tagc − ctag + tcag − ctga + tcga + gtac⎦, tgca − atcg + actg − cgta + tacg − gtca + ctag + gcta − tcag + ctga − catg − tcga

(5.14)

⎡

⎤ −gact + cgat − cagt + gcat − cgta + gtca − ctag − gcta + gatc + ctga + catg − gtac ⎢ ⎥ ⎣ agct − gact − tgca + acgt − atcg − cagt − atgc + tacg + gtca + tagc + ctga − tcga⎦, (5.15) atcg − agtc + atgc − actg + tgac − tacg − tagc − ctag + tcag + gatc + catg − gtac ⎡

⎤ −gact + tgca + gcat − tgac − tacg + gtca + tagc − gcta + tcag + gatc − tcga − gtac ⎢ ⎥ ⎣ acgt − agtc − cagt − atgc + actg + cgta + tagc − gcta + gatc + ctga − catg − tcga⎦. (5.16) agct − gact − cgat − acgt + atcg + cagt + gcat − actg − tacg − ctag + tcag + catg

indicated, respectively, by Ir , Ir , Ir . To outline the interpretation of these subspaces, the following notation is introduced: indicate by ±uk the sum of (the data indexed by) the permutations with the symbol u in the kth position that differ only by cyclically moving the remaining symbols in one direction (+) and in the opposite direction (−). Then, for example, the ﬁrst compoγ nent I1 = agct+gact−acgt−atcg−agtc−gcat+atgc+actg−gtca+gcta−gatc+gtac γ γ of Iγ can be represented as ±a1 ± g1 . It then follows that I1 = ±a1 ± g1 , I2 = γ

γ

±a3 ± g3 , I3 = ±a4 ± g4 . Similarly, I1

γ I3

=

γ ±c4 ± g4 , I1

=

γ ±t1 ± g1 , I2

γ

= ±c1 ± g1 , I2

= ±t3 ± g3 , and

γ I3

= ±c3 ± g3 ,

= ±t4 ± g4 .

582

M. Viana

Interpretation Keeping in mind that in the generating permutation the symbol g occupies position number 3, it is observed that each subspace compares g against the cyclic (eventually chiral) changes of the remaining symbols in the remaining positions, one subspace for each one of the three positions 1, 2, or 4 to which g is moved to. The joint reduction of the data space The vectors {F1 , ISgn , Iη , Iη , Iβ , Iβ , Iβ , Iγ , Iγ , Iγ } deﬁne the (rows of the) matrix B shown in Appendix A, relative to which BP1 B−1 = Diag(1, 0, . . ., 0), BPSgn B−1 = Diag(0, 1, 0, . . ., 0), BPη B−1 = Diag(0, 0, I4 , 0, . . ., 0), BPβ B−1 = Diag(0, . . ., 0, I9 , 0, . . ., 0), and BPγ B−1 = Diag(0, . . ., 0, I9 ). The regular action of S4 extended to the components of the joint basis gives a representation isomorphic to Diag(1, Sgnτ , η, ητ , βτ , βτ , βτ , γτ , γτ , γτ ). 6. Comments and summary This chapter developed the regular invariants for S3 and S4 and obtained their interpretations within the context of data indexed by these two groups. These regular invariants are to be understood in analogy to the usual invariants (leading to the sample mean and variance) that are obtained under the action of the full symmetric group shufﬂing the sample labels {1, 2, . . ., n}. The regular action shufﬂes the experimental labels (permutations) by multiplication and introduces a degree of arbitrariness. The interpretation of the regular invariants, in data analytic applications, should be better understood so that the corresponding parametric hypotheses can be formulated and scientiﬁc explanations suggested. It is also observed that symmetry studies of regular (hence transitive) actions are simpler to interpret. That is, in the regular case, Step (4) described in Section 3 always gives a single orbit. Otherwise, when orbits appear, as illustrated by (3.3), the canonical projections collect together pieces from different orbits and label the resulting classes by the irreducible representations, in one-to-one correspondence with the canonical projections. This “crossing out into nonisomorphic subspaces” clearly complicates any eventual interpretations. This is why the representations might be restricted to the underlying (transitive) orbits, for data analytic purposes.

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ B=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡

1 1 1 0 0 1 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1

1 −1 −1 0 −1 1 −1 0 0 1 1 0 1 0 1 1 0 0 −1 −1 0 −1 0 −1

1 −1 1 −1 0 −1 0 −1 0 −1 0 0 0 0 1 0 1 0 1 0 0 0 0 −1

1 −1 0 1 1 0 0 0 −1 0 1 0 −1 0 0 0 0 1 0 −1 0 1 0 0

1 −1 0 1 1 0 1 1 0 0 −1 0 0 −1 1 −1 −1 0 0 1 0 0 1 −1

1 −1 1 −1 0 −1 1 0 1 0 1 −1 0 0 −1 −1 0 −1 0 −1 1 0 0 1

1 −1 −1 0 −1 1 1 0 0 0 0 1 0 1 0 −1 0 0 0 0 −1 0 −1 0

1 1 −1 1 −1 0 0 1 0 −1 −1 0 0 −1 1 0 1 0 −1 −1 0 0 −1 1

1 1 0 −1 1 −1 −1 −1 0 1 0 0 1 0 1 −1 −1 0 1 0 0 1 0 1

1 1 0 −1 1 −1 1 1 0 0 −1 1 0 −1 0 1 1 0 0 −1 1 0 −1 0

1 1 −1 1 −1 0 1 0 1 0 0 −1 0 1 −1 1 0 1 0 0 −1 0 1 −1

1 1 −1 1 −1 0 0 −1 0 0 0 1 −1 0 0 0 −1 0 0 0 1 −1 0 0

1 1 0 −1 1 −1 0 0 −1 −1 0 0 0 1 0 0 0 −1 −1 0 0 0 1 0

1 1 0 −1 1 −1 0 0 1 0 1 −1 −1 0 −1 0 0 1 0 1 −1 −1 0 −1

1 1 −1 1 −1 0 −1 0 −1 1 1 0 1 0 0 −1 0 −1 1 1 0 1 0 0

1 −1 1 −1 0 −1 0 1 0 0 −1 1 −1 −1 0 0 −1 0 0 1 −1 1 1 0

1 1 1 0 0 1 0 −1 1 −1 0 −1 0 0 −1 0 −1 1 −1 0 −1 0 0 −1

1 −1 1 −1 0 −1 −1 0 −1 1 0 0 1 1 0 1 0 1 −1 0 0 −1 −1 0

1 −1 −1 0 −1 1 0 −1 1 0 0 −1 −1 0 −1 0 1 −1 0 0 1 1 0 1

1 1 1 0 0 1 −1 0 0 1 0 1 1 1 0 −1 0 0 1 0 1 1 1 0

1 −1 −1 0 −1 1 0 1 −1 −1 −1 0 0 −1 0 0 −1 1 1 1 0 0 1 0

1 −1 0 1 1 0 0 0 1 −1 0 −1 0 1 −1 0 0 −1 1 0 1 0 −1 1

1 1 1 0 0 1 0 1 −1 0 −1 0 −1 −1 0 0 1 −1 0 −1 0 −1 −1 0

1 −1 0 1 1 0 −1 −1 0 1 0 1 1 0 0 1 1 0 −1 0 −1 −1 0 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

Canonical Decompositions and Invariants for Data Analysis

Appendix A 583

584

M. Viana

References [1] K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, 2nd ed., Cambridge University Press, New York, NY, 2002. [2] H. Bacry, Leçons sur la theorie des groupes et les symétries des particules elémentaires, Dunot, Paris, France, 1967. [3] M. Viana, Symmetry studies and decompositions of entropy, Entropy 8, no. 2, (2006) 88–109. [4] M. Viana, Symmetry Studies- An Introduction to the Analysis of Structured Data in Applications, Cambridge University Press, New York, NY, 2008. [5] J-P. Serre, Linear Representations of Finite Groups, Springer-Verlag, New York, 1977. [6] P. Diaconis, A generalization of spectral analysis with application to ranked data, Ann. Stat. 17 (1989) 949–979.

Index β-rings, 451 λ-rings, 409 – Adams operation and, 418 – binomial rings v., 431 – deﬁnition of, 410–411 – morphisms of, 411, 423 – nontrivial example of, 416 – on one generator, 328 – on QSymm, 422 – σ-rings v., 411–412 – structure of, 416–417 – on Symm, 421–422, 425 – universal, one generator, 423–424 – Witt vectors and, 415 × √L -Hopf algebras, 230–231 Morita-equivalent, 195 μP , 375–376 μS , 375–376 π(P), 88 πX, 85 ϕ endomorphism, 399 ρX∗ , 104–106 functor, 402 ψK operation, 408 ×R – coalgebra, 185 – Hopf algebra, 191

-rings, 417–418 – Adams operation and, 418 – noncommutative, 418 ψ-twisted convolution algebra, 253 R ⊗k Rop , 207

A A-coring, 152. See also B|A-corings – comodules of, 261 – trivial, 246 A–D–E classiﬁcations, 512 A#D-modules, 162 Ae -rings, 286

A-linearization, 133 A-module – C-ferential, 149–150 – k-form of, 129 AS D-module algebras, 129 – Birkhoff-Witt, 162 – commutative, 161 – ﬁnitely generated, 169 – L/K extension and, 165, 169–170 – PV extensions of, 164–168 Abelian – arbitrary category, 165 – category – – simplicity criterion for objects in, 165–166 – extensions – – of ﬁeld p, 344–346 – factors, 147 – group – – graded, 378 – groups (A) – – multiplication on, 350–351 – tensor subcategory, 156 AbGroup, 349 abstract algebra, 70–71 action picture, 198 Adams, J. F., 92, 412 – ﬁrst Frobenius theorem, 412 Adams operation, 412–413 – -rings/λ-rings and, 418 adjoint pair, 241 adjunctions – coinvariants and, 257–258 – comodules and, 256–257 – counit/unit of, 241 admissible ideal, 527 afﬁne algebra, 53 – Hermetian/quadratic forms over, 37 – Lioville group scheme, 146 afﬁne group scheme (G), 128–129, 137–138, 146–147

585

586

Akian, Marianne, 31 Albert forms, 56–57 algebra. See also AS D-module algebras; bialgebroids; Cartier algebra; C-rings; central simple algebras; coalgebra; D-module algebras; depth 2, algebra extensions; Hopf algebra; Hopf algebroids; Kac-Moody algebra; K-algebra; Lie algebras; max-algebra; min-algebra; positive selfadjoint Hopf (PSH) algebras; Ringel-Hall algebras; Z(p) -algebras – abstract, 70–71 – afﬁne, 53 – base, 286 – Boolean, 4 – canonical, 511 – central simple, 47 – C-ferential, 149–152, 158 – – simple, 151 – Clifford, 70–71 – cofree, 158 – – over module, universality, 379 – of coinvariants – – of coring morphism, 260–261 – comodule, 133 – – module coalgebras and, 288 – concealed, 557 – corings and base extension, 246 – coseparable corings and ﬁrm, 270 – crossed differential, 101 – crossed modules, examples of, 87 – cubes and, 85 – data, 93 – depth 2 extension of, 203–204 – Dieudonné, 407 – differential graded – – corings and semi-free, 249 – – group-like elements and, 258–259 – – semi-free, 249 – double, 219 – extension, 252, 283–285 – – balanced, 203 – – equivalence, 203–204 – ﬁrm, 248 – free – – Symm and, 378 – – universality property of, 378 – Frobenius, 273–274 – generic composition, 538 – Hall, 509

Index

– hereditary, 526 – inverse to subdivision, 90, 94, 101, 106 – module, 198 – necklace, 429–430 – over monad, 243 – path – – of quivers, 525–528 – – quotient of, 509 – plactic, 491 – quasi-hereditary, 510–511 – quaternion, 44 – regular, 248 – representation of, 299 – representations of ﬁnite-dimensional, 509 – right comodule, 212–213 – rings over, 252 – Schur, 510 – separable – – map, 270 – splitting, 168 – – minimal, 169 – stratiﬁed, 510–511 – of symmetric functions, 452–454 – tabular type, 511 – tensor, 378 – tensor module, 150 – thin element, 108 – tropical, 5 – tubular, 557 – twisted convolution, 293 – Witt, 329 – ψ-twisted convolution, 253 algebraic geometry, 509 algebraic quantum torus, 207 almost split sequence, 528 alternating invariant, 574–575 Amano, K., 127–130 Amitsur cohomology, 170 André, Y., 129 anisotropic quadratic form, 38 antipode, 204, 288 – on graded bialgebras, 365 Arason, J., 40, 45, 47 Arason-Pﬁster Hauptsatz, 40, 48, 53–55 A-ring – product in, 252 – unit in, 252 Arnold, V. I., 512 arrows, 239 Artin, E., 36, 60 – Hilbert problems, solution of, 70

Index

Artin-Hasse exponential, 415–416 Ashley, N., 116 Atiyah, M. F., 436, 511 Auslander, M., 509 Auslander-Reiten – quivers, 529–530, 534–535, 546 – sequence, 528–529 – theory, 528 – translations, 529 autoduality – of Symm, 374 – of SymmR , 399 *-autonomous structure, 232

B Bacry, H., 566, 573 Baeza, R., 53 Bajnok, D., 232 balanaced extension, 203 Bálint, I., 232 Balmer, P., 64 Bangor thesis of Al-Agl, 105 base changing morphisms, 186 Baues, H.-J., 112 Baumann, P., 549 B-comodules, 197–198 B-modules, 193–194 B|A-corings, 253 Beasley, LeRoy, 31 Beck, J., 549 Becker, E., 68 Beck theorem, 280 Bence, S. J., 566 Bernstein, I. N., 531 Bernstein, J., 439 Bernstein-Gelfand-Ponomarev (BGP) reﬂective functors, 535–536 Bernstein morphism, 439–440 best possible bounds, 22 bialgebras, 186 – Galois extensions and, 202–203 – – non-uniqueness of, 202 – graded – – antipodes on, 365 – Hopf algebra and, 288 – weak, 186–187, 291 – – Frobenius separable, 187 bialgebroids, 286–288 – action picture and, 198 – coaction picture and, 199–200

587

– cocycle double twists, 189–190 – Connes-Moscovici, 192 – Drinfel’d double, 191 – Drinfel’d twist, 189 – duality of, 188–189 – examples, 186–188 – Galois theory of, 220 – Hopf algebroid comodules and, 209–211 – key properties, 175 – left, 184–186 – L-rings measurement by, 223 – monoidality of, 231 – Morita base change of, 191–192 – morphisms, 185–186 – quasi-triangular, 196 – right, 183–186 – Scalar extension of, 193 – study/history of, 182 bicategory, 308 bicoalgebroids – C-rings and, 309 – over coalgebra, 197 bicomodule – cleft, 286 – regular, 256 bicrystals, 491 – isomorphism of, 497 – M , 500–501 bideterminant, 6 bi-Galois extensions, 200–202 bilinear – forms – – Witt rings of, 39–40 – space, 37 binomial rings – adjoints of, 433 – deﬁnition of, 430 – free, 430–431 – λ-rings v., 431 Birkhoff-Witt – coalgebra, 158 – AS D-module algebras, 162 – T(C+ ), 158 Bõhm, G., 201, 204, 206–207, 218 Boolean – algebra, 4 – semiring, 5 – – bounds and, 23, 26–28 – spaces, 41, 68 Borsuk-Ulam theorem, 58 bounded latice, 4

588

bounds – antinegative semiring and, 25–26, 28–29 – arbitrary semiring and, 24, 27 – best possible, 22 – Boolean semiring and, 23, 26–28 – column rank and, 25 – exact, 22 – row rank and, 25 bow-tie diagram, 250 braided category, 182 – monoidal, 196 Brauer equivalence, 347 Brauer group – central simple algebras and, 347 – for corings, 310 – quaternion algebra of, 43 Breen, L., 114 Bröcker, L., 66, 68 Brown, R., 85, 89 Brzezinski, T., 182, 201, 232 BU – classifying space, 327 Burnside ring, 442–443. See also Witt-Burnside functors – deﬁnition of, 444 ˆ – Frobenius on B(Z), 449 ˆ – ghost component morphism of B(Z), 446 ˆ – Verschiebung on B(Z), 449 Burnside theorem, 444 Burroughs, J., 436 Butkovic, Peter, 31 ˆ B(Z)ψ – Frobenius on Burnside ring, 449 – ghost component morphism of Burnside ring, 446 – Verschiebung on Burnside ring, 449

C canonical – algebra, 511 – decomposition theorem, 569–570 – dual bases, 247 – entwining, 295 – maps, 198–199 – – lifted, 297 – – surjectivity of, 224 – projections – – for S3 regular invariants, 572–573 – quantum group and, 510 Carrasco, P., 116

Index

Cartan – datum, 554 – – symmetric/irreducible, 514 – form, 532 – matrix, 475 – – elliptic, 556 – – generalized, 517 – – graph of symmetric, 514 – – indecomposable, 516–517 – – of quiver, 532 – – symmetrizable, 514–515 Cartier, P., 348, 352, 362, 397 Cartier algebra, 401–404 Cartier ﬁrst theorem, 371 Cassels, J. W. S., 60 categorical descent theory, 279–280 category of modules, 240 category theory, 239–244 C-bicoalgebroid, 309 CRing, 325, 345, 349 C-rings, 304–305 – bicoalgebroids and, 309 – coendomorphism, 308–309 – dual-Sweedler, 306–307 – entwined structures and, 307 – Galois, 309 – matrix, 308 – modules over, 305–306 – separable/split, 306 CW -complexes, 100 Cegarra, A. M., 116 centralizing conditions, 196 central simple algebras, 346–347 – Brauer group and, 347 – crossed product and, 347–348 C-ferential – algebra, 149–152 – – integral domain, 158 – – simple, 151 – A-module, 149–150 – extension – – exponential element in, 154 – – ﬁnitely generated, 155 – – primitive element in, 154 – – splitting ﬁelds for, 155 – ﬁeld, 150 – – minimal splitting, 155 – Galois correspondence, 152–153 – K-algebra, 150–151 – L/K extension, 155 – – primitive, 154

Index

– module, 149 – – constants in, 150 – morphism, 149 – PV extension, 152–153 – splitting ﬁeld, 155 chain. See also Liouville normal chain – complexes – – with groupoids of operators, 114–116 – semiring, 5 – syzygies, 88 chain of syzygies, 101 characters, 306 charge of w, 498 Chari, V., 549 Chern class, 409 Chevalley, C., 330 Chevalley generators, 515, 548 Choi, M. D., 60 Clarence Wilkerson theorem, 414–415 class numbers, 50 cleft extensions, 221–224 Clifford algebra, 70–71 Cline, E., 510 coaction – image of, 255 – left, 255 – right, 254 coaction picture, 198 – bialgebroids and, 199–200 – Galois extension and, 200 coalgebra, 185, 242–243, 246 – bicoalgebra over, 197 – Birkhoff-Witt, 158 – cofree, 379–381 – – Symm/NSymm/QSymm and, 381 – comodule algebra and module, 288 – comodule as, 256 – of comonad, 242 – Galois comodule, 296 – Galois extensions, 295 – H-module, 292 – pointed irreducible, 153 – principle bundles, 295 – smooth cocommutative, 162 – tensor, 378 coassociative law, 244 coassociativity, 244 – of Sweedler-Heyneman notation, 245 cocycle double twist, 189–190 coendomorphism context, 309

589

cofree – algebra over module – – universality of, 379 – coalgebra, 379–381 – – Symm/NSymm/QSymm and, 381 – Witt vectors, 420–421 co-Frobenius corings, 274 cofunctor, 284 Cohen – functor, 343–344 – rings, 342–343 – – of k, 344 cohomology, 45 coideals, 245 coinvariants. See also g-coinvariants – adjunctions, 257–258 – algebra of – – of coring morphism, 260–261 – of comodule, 257 – coring morphism and, 260–261 – cotensor product and, 259–260 cokernals, 263 collapsing – elementary, 106 – homotopy and, 107 – partial boxes, 108 Colliot-Thélène, J. L., 61 column rank, 15–16 – bounds and, 25 – spanning, maximal, enveloping, 16, 19 comatrix – corings – – comodules and, 261–262 – – contexts, 247–248 – – dual ring of, 253 – – ﬁnite, 247 – – Frobenius bimodules and, 272–273 – – inﬁnite, 247–249 – – separable bimodules and, 271 combinatorial ranks, 17 comodule, 242–243. See also bicomodule – adjunctions of, 256–257 – algebra, 133 – – Hopf algebroid, 213 – – module coalgebras and, 288 – category of, 263–268 – of C-bicoalgebroid, 309 – as coalgebra, 256 – coinvariants of, 257 – cokernels for, 263 – comatrix corings and, 261–262

590

– coproduct and, 264 – A-coring and, 261 – corings and – – ﬂat, 264 – – Hopf module as, 292–293 – – locally projective, 266 – descent data and, 261 – dual rings and, 265 – with ﬂat connections, 306 – Galois, 275 – – A as, 275–276 – – coalgebra, 296 – – connections, 276 – – Morita contexts and, 286 – – principle, 278–279 – – strong/weak structure theorem, 277 – – structure theorem, 276–277, 283 – Grothendieck category, 264–265 – injective, 268–269 – kernals for, 264 – k-linear category, 263 – left, 255 – monoidal structure of, 287–288 – Morita context and, 282–283 – morphisms of – – factorizable, 256 – as noncommutative space, 297–298 – right, 254 – semisimple, 277 – simple, 277 – structure of, 268 – Sweedler-Heyneman notation and, 255 – weak, 302–303 – Wisbauer σ-category, 267 comonadic triangle, 279–281 comonads, 418–419 – adjoint functors and, 243 – coalgebra/comodule of, 242–243 – corings and, 245 – Witt vector, structure of, 419–420 comonoid, 177 – counit of, 177 complicial set, 118 comultiplication, 177 – cut, 373 – inner product v. product, 368 – product, 366–367 – on QSymm, 374–375 – – co-unit for, 376–377 – Symm morphisms and, 367–368 connection pair, 95

Index

connections – ﬂat, 262 – Galois comodule, 276 Connes, A., 98, 224 Connes-Moscovici bialgebroids, 192 conservative extension, 554 contramodules, 310 Coombes, K. R., 410 coproduct – comodules and, 264 – of coring, 244 – counitality of, 244 Cordes, C. M., 50, 67 corings, 131. See also A-coring; co-Frobenius corings; L-coring; R-corings; S|R-corings – algebra and – – semi-free differential graded, 249 – base algebra extension, 246 – Brauer group for, 310 – category of, 246 – comatrix – – comodules and, 261–262 – – contexts of, 247–248 – – dual ring of, 253 – – ﬁnite, 247 – – Frobenius bimodules and, 272–273 – – inﬁnite, 247–249 – – separable bimodules and, 271 – comodules and, 264 – – Hopf module as, 292–293 – comonads and, 245 – co-opposite, 179 – coproduct of, 244 – cosemisimple, 277–278 – – Jacobson-Bourbaki correspondence and, 278 – coseparable, 269 – – as ﬁrm algebra, 270 – cosplit, 270 – counit of, 244 – deﬁnition, 244 – dual, 253–254 – entwined structures and, 251, 262 – – weak, 251–252, 302 – extension, 283–285 – – Morita context for, 285–286 – Frobenius, 271–272 – – generalizations of, 274–275 – Galois, 275 – – Hopf-Galois extension as, 295–296

Index

– history/study of, 238 – integral in, 270 – lax, 301, 303 – – structure/usage, 304 – locally projective – – comodules of, 266 – module properties of, 268 – morphisms of, 245 – – algebra of coinvariants of, 260–261 – – coinvariants of, 260–261 – Picard group for, 310 – prime/coprime/cosemiprime, 310 – representable monoidal functors and, 253 – semisimple, 269 – simple, 277–278 – Sweedler, 246 – – dual ring of, 253 – weak, 301–302 – – entwined structures and, 251–252, 302 cotensor products, 197, 210 – coinvariants and, 259–260 counits, 177, 244 – of corings, 244 covers – equivalent morphisms of, 299 – ﬁnite, 298 – ﬂat, 300 – space, 299 Coxeter functors, 536 Craven, T. C., 41, 62, 66 crossed – complexes, 85 – – advantages of, 100–101 – – bimorphism of, 110–111 – – categories of, 102–103 – – free basis of, 112–113 – – fundamental, 99–100 – – simplicial groups and, 116–117 – – tensor product of, 111–112 – differential algebra, 101 – modules – – algebraic examples, 87 – – calculation of, 88–89 – – of groupoids, 100 – – homotopy and, 88 – – identities among relations, 88 – product, 224 – – central simple algebras and, 347–348 – set, 114 Crowell, I., 115 crystal. See also bicrystals

591

– abstract – – deﬁnition of, 480 – base, 474, 510, 539 – – tensor product of, 478–479 – – theories of, 476–478 – direct sum of, 480 – dual, 484 – equivalence, 481 – graph, 474 – – monomial/path/tableaux realization of, 479 – – structure of, 477 – isomorphism, 480, 490 – semi-normal, 480–481 – sub-, 481 – version of Robinson-Schensted correspondence, 489 – of words, 481–482 – of Young tableaux, 482–484 – – rational, 484–486 cubes – algebra of, 85 – commutative, 95, 101, 105 – in double categories, 95–96 – higher homotopy groupoids and, 85 – opposite forces in n-, 107 – resurgence of use of, 102 cubical classifying space, 101–102 – ﬁltered, 99 cubical nerve, 102 cubical sets, 99 curvature of connection, 262 curves – functor, 370 – in Hopf algebra, 370 – NSymm and, 370–371 – SymmR , 398 cut comultiplication, 373 cyclic sets – almost ﬁnite, 445 – induction for, 447 cyclotomic identity, 429 cylinder object, 101 Czogala, A., 52

D D2 quasi-bases, 250 Dai, Z. D., 58, 60 Dai-Lam-Pang theorem, 58–59 Damiani, I., 547

592

data – algebraic, 93 – descent – – category of, 280 – – comodule and, 261 – topological, 93 Date, E., 489 Day, B., 186 DE – matrix – – of SymmR , 399–400 – principle, 400–401 D-module algebras, 129. See also AS D-module algebras – invariants, 160 – principle, 164 – simple, 161 D-module ﬁeld, 163 Dedekind – domains – – Witt ring of, 63–64 – rings, 62 degeneracies, 94 – double, 94 DeMeyer, F. R., 52 depth 2 – algebra extensions, 203–204 – extensions – – K-algebra and, 250 – Frobenius extensions, 221 – quasi-bases, 203, 250 De Rham theory, 119 derived category, translation of, 551 descent – comonadic triangle and, 279–281 – data – – category of, 280 – – comodule and, 261 – Galois, theory of, 214 – monads of, 280 – theory, categorical, 279–280 descent theorem, 214 determinant, 30–31 – matrix and, 5–7 – singularity and, 7–10 devisor theory, semigroup with, 53 Diaconis, P., 572, 576 Dickson, L. E., 71 Dieudonné, J., 362 Dieudonné algebra, 407 difference

Index

– equation, 159–160 – ﬁeld, 159 – – Fibonacci recurrence equation and, 160 – – inverse, 159 dimension, 13 – of group scheme, 145 – vector representation of quiver, 521 dimension index, 40 dimension vector, 521, 552 – M, 530–531 distributive lattice, 4 divided power – in Hopf algebra, 370 – sequences, 370 – subalgebra, 519 divided power sequence, 150 Dlab, V., 509, 539 Doi, Y., 221, 226 Doi-Koppinen Hopf modules, 290–292 Dold, A., 362 dominant integral weight, 475 domination, 14 Doran, W., 359 double algebras, 219 double categories – connection pair on, 95 – cubes in, 95–96 – edge symmetric, 93–94 – morphism of, 94 Dress, A., 443 Dress local global principle, 62 Drinfel’d, V. G., 474, 517 Drinfel’d double, 191 Drinfel’d twist, 189 Drozd, Y. A., 238 dual bases canonical element, 247 duality. See also autoduality – of bialgebroids, 188–189 – Galois R-corings and, 181–182 – Galois R-rings and, 181–182 dual matrix invariant, 18 dual weight lattice, 475 Dugger, D., 59 Dwork, B., 362 dynamical system theory, 362 Dynkin diagrams, 512, 531

E Eckmann, B., 86 ETC, 66–67

Index

Ehlers, P. J., 116 Ehresmann, C., 91–92 Eilenberg-Mac Lane, 103, 113 – theorem, 85, 101 elementary operations, 65 Ellison, W. J., 60 Elman, R., 37, 47, 56, 57 endomorphism – C-rings and co-, 308–309 – dual rings as natural, 252 – Frobenius – – of QSymm, 386–388 – Frobenius compared to Verschiebung, 391 – Hopf – – of Symm, (h – h)-matrix, 398 – Hopf algebra – – Symm, matrix of, 398 – – of WH (p) , 405–407 – of inﬁnite dimensional additive group, 404–405 – ring of Hopf – – of Symm, 392–393 – Witt vector as, 402–404 entwined – canonical, 295 – map, 251 – – Galois extension and, 295 – modules, 238, 262–263 – – weak, 263 – structures, 238, 250 – – cleft, 286 – – corings and, 251 – – corings and weak, 251–252 – – C-rings and, 307 – – lax, 301, 303–304 – – mixed distributive law and, 251 – – partial, 301, 303–304 – – self-duality of, 293 – – weak corings and, 251–252, 302 epimorphism, 240 equivalence of category, 242 equivariant projective module, 229 Euclidean diagrams, 517 Euler – form, 531–532 – – symmetric, 532 – quadratic form, 531 Euler, L., 36 exact bound, 22 exceptional -model, 554 excision theorem, 104

593

existence lemma for exterior power structures, 413 exponential element, 134 exterior powers, 449 – existence lemma for, 412

F face maps, 93 factor rank, 15 Faddev-Reshetikhin-Takhtajan construction, 193 fans, 68 Faria Martins, J., 113 Fermat, P., 36, 69 Fermionic Fock space, 498 Feynam identity, 428 Fibonacci recurrence equation, 160 ﬁbration theorem, 106–108 ﬁelds. See also function ﬁelds; global ﬁelds; Pythagorean ﬁelds – Abelian extensions of p, 344–346 – C-ferential, 150 – difference, 159 – – Fibonacci recurrence equation and, 160 – – inverse, 159 – differential, 130–131 – – Liouville extension of, 145 – – L/K extension and, 140, 142 – – splitting extension of, 128 – – subﬁeld of constants of, 128 – D-module, 163 – extension, excellent, 44 – formally real, 41 – global – – Hasse principle for quadratic forms over, 51 – level of, 36, 52 – minimal splitting, 141–142 – – unique existence of, 143–144 – non-formally real, 39 – nonreal, 56 – number – – quadratic form over, 48 – p-adic Witt vectors over, 342 – preordering, 67–68 – radical, 67 – skew – – Witt rings and, 64 – splitting, 139–143 – – C-ferential, 155 – – of C-ferential extensions, 155

594

– – characterization of, 141–142 – – inﬁnite dimensional, 142 – – minimal, 158–159 – – minimal, C-ferential, 155 – square class number, 50–51 ﬁltered – homotopy, 109 – space, 99 – – connected, 103 ﬁnite etale, 146 ﬁrm module, 248 – right, 270 – strong structure theorem on, 281 ﬁxed point index, 362–363 Fock space representations, 497–502 forgetful functor, 241 – inverse, 211 – monoidal, 199 – obtaining, 196–197 – relative separability of, 226 formal groups, 361 Fourier transforms, 570 Fox, R. H., 114–115 Franke, C. H., 128 free – actions – – principal bundles and, 293–294 – algebras – – Symm and, 378 – – universality property of, 378 – binomial rings, 430–431 – crossed complexes, 112–113 – crossed resolution, 88, 112–113 – differential calculus, 114 – resolution, 88 Frobenius. See also co-Frobenius corings – algebra, 273–274 – bimodules, 272 – – comatrix corings and, 272–273 ˆ – on Burnside ring B(Z), 449 – corings, 271–272 – – generalizations of, 274–275 – depth 2 extensions, 221 – element, 249 – endomorphisms – – of QSymm, 386–388 – – Verschiebung endomorphism compared to, 391 – extension, 249, 272 – – corings and, 249–250 – – towers of, 273

Index

– functional, 221 – functors, 272 – homomorphism, 249 – Hopf algebroids, 208, 218–219 – on necklace rings, 431–432 – operations, 415, 449 – – ghost component characterization of, 384 – – p-adic Witt vectors and, 335–336 – – Symm and, 383–386 – rings, 21, 218–220 – separable, 187 – system, 272 Frobenius, G. F., 434, 436 Frobenius theorem – ﬁrst Adams, 412 full subcategory, 239 functional equation integrality lemma, 360 function ﬁelds, 36 – global, 51–52 – quadratic forms and, 55–56 – selected applications, 53–56 functors. See also Bernstein-Gelfand-Ponomarev (BGP) reﬂective functors; cofunctor; forgetful functor; functor; Witt-Burnside functors – additive, 240 – adjoint, 241–242 – – comonads and, 243 – – monads and, 243 – of big Witt vectors, 348–349 – category of, 241 – Cohen, 343–344 – contravariant, 240 – coseparable, 271 – covariant, 240 – Coxeter, 536 – curve, 370 – faithful, 240–241 – Frobenius, 272 – full, 240–241 – Green, 442 – induction, 241 – linear, 240 – monoidal, 244 – – B|A-corings and, 253 – – corings and representable, 253 – as multiplication, 244 – of one power series, 349–351 – of p-adic Witt vectors, 331–333 – rational, 266–267

Index

– separable, 271 – similar, 274 – tensor, 241 fundamental ideals, 40 fusion category, 206

G Gabriel, P., 509, 531 Gabriel theorem, 514, 533–536 Gale-Ryser theorem, 375, 440–441 Galois. See also bi-Galois extensions; Hopf-Galois extensions – bialgebroid theory, 220 – coalgebra comodule, 296 – cohomology, 45 – comodules, 275 – – A as, 275–276 – – connections, 276 – – Morita contexts and, 286 – – principal, 278–279 – – strong/weak structure theorem for, 277 – – structure theorem, 276–277, 283 – corings, 275 – – Hopf-Galois extension as, 295–296 – correspondence, 128 – – C-ferential, 152–153 – – characterization, 153–158 – – for L/K extension, 135–139 – – theorem, 166 – C-rings, 309 – descent theory, 214 – extension – – bialgebroids and, 202–203 – – coaction picture of, 200 – – coalgebra, 295 – – cyclic, 344–346 – – entwined map and, 295 – – Hopf algebra, structure of, 224–229 – – Hopf algebra and, 220–221 – – noncommutative ring, 300–301 – – non-uniqueness of bialgebra of, 202 – groupoids, 92 – Hopf, extensions, 174 – property, 198 – R-corings, 180–181 – – duality and, 181–182 – R-rings – – duality, 181–182 – – with respect to character, 179 Gaubert, Stephane, 31

595

Gauss, C., 36 Geigle, W., 511, 557 Gelfand, I. M., 370, 531. See also Bernstein-Gelfand-Ponomarev (BGP) reﬂective functors generic splitting towers, 55 geometric realization, 114 geometry – algebraic, 509 – noncommutative algebraic, 297 – noncommutative differential, 293 g-coinvariants, 259 G-modules, 88 G-sets – almost ﬁnite, 444 – induction/restriction for, 443–444 – symmetric powers of, 448–449 G-spaces, 450 ghost component – equations, 334–336 – formalism, 390–391 – Frobenius operation, characterization of, 384 – integrality lemma, 361 – morphism, 446 ˆ – – of Burnside ring B(Z), 446 – Verschiebung operation, characterization of, 382 Glazebrook, J., 119 GLn primitive matrix, 155 global ﬁelds – Witt equivalence of, 51–53 – Witt rings and, 52 globular – methods, 119 – set, 117 Godement, R., 418 Golan, Jonathan, 31 Gondran, M., 11, 20 Gondran-Minoux dependent, 11 Grassman manifolds, 440 Green, J. A., 509, 538 Green functor, 442 Grothendieck – category – – spaces as, 297 – comodule category, 264–265 – group, 531 – ring, 445 Grothendieck, A., 410 group-like elements, 258–259 – coinvariants with respect to, 180

596

groupoids. See also higher homotopy groupoid; van Kampen theorem; w-groupoids – chain complexes with, 114–116 – commutative squares and, 91–92 – crossed module of, 100 – double, 93, 97 – fundamental – – π1 (X, x0 ), 89–90 – – precision for, 90 – – on set of base points, 89 – Galois, 92 – symmetry, 174 group scheme – dimension of, 145 – ﬁnite etale/connected, 146 – Lioville afﬁne algebra, 146 Grunspan, C., 200

H Hajac, P. M., 229 HAL. See homotopy addition lemma Hall. See also Ringel-Hall algebras – algebras, 509 – inner product, 435 – polynomial, 537 – set, 428 Hall, P., 509 Happel, D., 551, 558 Harrison – criterion, 49 – sets, 68 Harrison, D. K., 49 Hasse. See also Artin-Hasse exponential – algebra, 43 – invariants, 43–44 – principle for quadratic forms over global ﬁelds, 51 Hasse, H., 36, 47, 69–71, 325, 330, 342 Hauptsatz, 40 Hausdorff space – free action of compact group on, 293–294 Hazewinkel, M., 360, 369, 379 HCL. See homotopy commutativity lemma Hermitian forms, 37 Hesselholt, L., 362 Heyworth, A., 113 H-comodule – isomorphism and, 211 – maps, 210 – right, 209

Index

HHvKT. See higher homotopy van Kampen theorem Higgins, P. J., 85, 89, 92, 98 higher homotopy groupoid – background/history, 86 – base points, fundamental, 89–92 – cubical, 85 – geometries of, 117 – main results for, 98–100 – researching, 92–98 – theorem, 90–92 higher homotopy van Kampen theorem (HHvKT), 85, 98 – calculating, 103–104 – multiple composition and, 105 – sketch proof of, 108–109 higher level orderings, 64 highest weight vector, 476 Hilbert, D., 69–70 Hilbert problems, 36, 69 – Artin solution of, 70 – 17th, 70 Hilbert-symbol equivalence (HSE), 51 – generalizations, 53 – tame, 52–53 – wild, 52 Hilton, P., 86 Hintze, S., 103 Hirata, K., 216 Hirzebruch polynomials, 407–409 – multiplicative, 409 Hobson, M. P., 566 Hobst, D., 201 Hoffman, D. W., 54, 60 homogeneous stable tube, 545 homomorphism, 44–45 – Frobenius, 249 – Milnor conjecture and, 46 – total residue, 63 homothety operators, 388 – Witt vectors and, 327 homotopy. See also higher homotopy groupoid; higher homotopy van Kampen theorem – colimits, 101 – collapsing and, 107 – crossed modules and, 88 – extension property, 107 – ﬁltered, 109 – local-to-global problems in, 85–86 – map classiﬁcation and, 113–114 – morphisms, 110

Index

– relative, 97 – simple, 86 homotopy addition lemma (HAL), 96 homotopy commutativity lemma (HCL), 96 Hopf. See also entwined, modules; positive selfadjoint Hopf (PSH) algebras; relative Hopf modules – bimonads, 186 – classiﬁcation theorem, 115–116 – condition, 59 – endomorphisms of Symm – – (h – h)-matrix of, 398 – – ring of, 392–393 – Galois extensions, 174 – ideal, 136 – – normal, 137 – module, 288–289 – – as comodule of corings, 292–293 – – Doi-Koppinen, 290–292 – – fundamental theorem of, 214 – – relative, 289–290 – – right-right relative, 199–200 – monads, 186, 194 – – left, 232 – – warning, 232 Hopf algebra, 143. See also Symm – bialgebras and, 288 – character on, 206 – commutative, 145 – curve in, 370 – divided power sequence in, 370 – endomorphisms of Symm – – matrix of, 398 – endomorphisms of WH (p) , 405–407 – of functions, 294 – internal symmetry of quantum models, 174 – invariants for, 161 – primitive in, 365 – PV and, 128–129, 153, 164 – PV L/K extension and, 134 – QSymm and, 373 – on quantum group Uq (gln ), 476 – ×R , 191 – R(S) ring and, 437 – symmetry in quantum ﬁeld theory, 174 – weak, 206–207 Hopf algebroids. See also ×L -Hopf algebras; Lu Hopf algebroid – axioms, 204–206 – basic properties of, 207–208 – cleft extensions by, 221–224

597

– comodule algebra, 213 – comodules, 175, 208–209 – – bialgebroids and, 209–211 – – coinvariants in, 211–212 – Frobenius, 208, 218–219 – fundamental theorem of, 214 – Galois extensions by, 220–221 – – structure of, 224–229 – integrals for, 175 – K-algebra and, 212 – Maschke-type theorems for, 216–218 – non-degenerate integral for, 219 – study/history of, 174–175 Hopf-Galois extensions – canonical entwining for, 295 – as Galois coring, 295–296 – principal bundles as, 294–295 horizontal identities, 94 Hornix, E. A. M., 57 HSE. See Hilbert-symbol equivalence Huebschmann, J., 88 Hurewicz, W., 85–86, 92, 98, 118 Hurwitz-Radin theorem, 59 hyperbolic plane, 38

I ideals – fundamental, 40 – prime, 41 identities among relations, 88 i-good signs, 482 imaginary root, 516 induction – for cyclic sets, 447 – for G-sets, 443–444 inﬁnite dimensional additive group, 404–405 inner product – Hall, 435 – product comultiplication v., 368 – spaces – – metabolic/similar, 61 – – Witt equivalence and, 61–62 – on Symm, 355 integrality lemma, 359–360 – functional equation for, 361–362 – – constructions for, 360–361 – – ingredients for, 360 integral theory, 214–216 interchange law, 94 inverse equivalences, 242

598

invertible, 9 involution, 47 Isaksen, D., 59 isometric – quadratic space, 37 isomorphisms – of bicrystals, 497 – crystal, 480, 490 – itp, 446 – natural, 241 – quadratic forms and, 46 – quasi-, 551 – retraction/section of, 240 – strong/strict, 244 – T , 446–448 isotropic quadratic form, 38 isotropy, 38–39 itp isomorphism, 446 Izhakian, Z., 11 Izhakian dependent, 11 Izhboldin, O. T., 55–57

J Jacobi-Trudy formulas, 356 Jacobson-Bourbaki correspondence, 278 Jannsen, U., 61 Jimbo, M., 474, 489, 517 Jordan classiﬁcation, 522 Jordan-Hölder multiplicity, 530

K K2 – symbols, 44–45 Kac, V., 509, 534, 551 Kac-Moody algebra, 474, 479 – construction of, 550–554 – derived algebra of, 514 – quantum groups and, 514–519 Kadison, L., 183, 188 Kahn, B., 56 Kan, D., 102 Kaplansky, I., 56–57, 67, 71, 343 Kapranov, M., 549 Kapranov rank, 18 Karpenko, N., 37, 54, 55–56 Kashiwara, M., 474, 476–477, 491, 510, 539 Kashiwara operators, 480, 493, 498 Kassel, C., 549 Kato conjecture, 61 Kazhdan, D., 510

Index

kernals, 263–264 k-form, of A-module, 129 K-algebra, 176 – C-ferential, 150–151 – depth-2 extension of, 250 – differential, 130–131 – Hopf algebroids and, 212 – maps, 131–132 – R-rings and, 177–178 – separable, 144 K-dimension, 156 k-linear category, 239 – comodules, 263 K-modules – differential, 139 – – cyclic, 140 – monoidal, 176 K-rank, 168 Kleiner, M., 238 Knebusch – degree, 55 – theorem, 55 – Witt group of a scheme, 64 Knebusch, M., 36, 55, 62, 65 Knebusch-Milnor exact sequence, 63 Knonecker, L., 523 Knuth equivalence, 486–488 Knuth transformation, 487 Knutson, D. E., 436, 495 König theorem, 18 Kontsevich, M., 238 Koprowski, P., 53 Kreimer, H. F., 133, 224 Krob, D., 370 Kronecker quiver, 523, 545–549 Krull dimensions, 142 Krull-Remak-Schmidt theorem, 521–522 Krüskemper, M., 48 Kula, M., 66–67 Kula theorem on Witt rings, 66–67 Kummer theory, 346 Kuratowski-Zorn lemma, 68

L Labesse, J. P., 114 Lam, T. Y., 37, 47, 54–55, 57–58, 60, 68. See also Dai-Lam-Pang theorem lambda ring, 328, 409 Larson, R. G., 218 Lascoux, A., 370, 491

Index

lattice – bounded, 4 – distributive, 4 – permutation, 489 Lazard, M., 332 Leclerc, B., 370 Leep theorem, 57 left – adjoint, 241 – almost split, 528 – basis, 12–13 – bialgebroids, 184–186 – character – – on R-rings, 178 – coaction, 255 – cointegral, 215 – comodule morphism – – of R-corings, 180 – comodules, 255 – dimension, 13 – Hopf monads, 232 – integral, 215 – invertible, 9 – linear combination, 10 – linearly independent, 11 – linear span, 10–11 – module – – for R-rings, 178 – semimodules, 10, 12 – singular, 8 – – matrix, 7 left-right symmetry, 255 Lens conference, 55 Lenzing, H., 511, 557 level, 58–59. See also Pﬁster Levelt, A. H. M., 143 Lewis, D. W., 59, 64 lextensive categories, 92 Lie algebras, 428 – classes of, 513–514 – elliptic, 555–558 Lin, Y., 510, 549, 555, 557–558 linear – category, 239 – combination, 10 – dependence, 11–13, 22 – differential equation, 140 – functor, 240 – independence, 11–13, 22 – maps – – equivalence of, 522

599

– span, 10–11 linearization, 133 linked quatrenionic mappings, 65 Liouville – afﬁne algebra group scheme, 146 – extension, 128 – – of differential ﬁelds, 145 – group schemes – – additive/multiplicative, 146 – L/K extension as, 145 Liouville normal chain (LNC), 146–147 Littelmann, P., 479, 491 Littelmann path model, 479 Littlewood-Richardson – rule, 435, 474, 486, 489–491 – tableaux, 490 Liulevicius, A., 436 Liulevicius theorem, 365–366 L-coring, 135 L/K extension, 128–130 – C-ferential, 155 – – primitive, 154 – differential ﬁelds, 140, 142 – AS D-module algebras and, 165, 169–170 – ﬁnitely generated, 169 – Galois correspondence for, 135–139 – as Liouville, 145 – primitive, 134–135 – principal differential ring for, 132–133 – PV group scheme for, 137–138 – PV Hopf algebra of, 134 – rings for – – simple differential, 136 LNC. See Liouville normal chain local-to-global – homotopy problems with, 85–86 – noncommutative theorems, 85 Loday, J. L., 118–119 loops, 519 lower bound, 5 L-rings, 223 Lu, J. H., 183 Lu Hopf algebroid, 230 Lusztig, G., 510, 539, 541 Lyndon words, 422, 428

M Macdonald, I. G., 436 Mackaay, M., 118 Mackey double coset theorem, 437

600

MacLane, S., 85, 87, 342 majorization ordering, 440 Mammone, P., 57 maps – afﬁne – – of noncommutative spaces, 297 – anchor, 184 – antipode, 204, 288 – canonical, 198–199 – – lifted, 296 – – lifting, 224 – – surjectivity of, 224 – characteristic, 435 – – in representation theory of symmetric groups, 435 – cointegral, 269 – colinear, 254 – cubical – – multiple composition determining, 106 – entwined, 251 – – Galois extension and, 295 – face, 93 – ﬁxed points of iterated, 362 – H-comodule, 210 – homotopy classiﬁcation of, 113–114 – iterated – – Witt vectors and ﬁxed point indices of, 362–363 – iterates of continuous, 362 – K-algebra, 131–132 – linear – – equivalence of, 522 – noncommutative afﬁne space and, 297 – reduced counit, 307 – separable algebra, 270 – source, 183 – target, 183 Marshall, M. A., 67 Maschke-type theorems, 216–218 Masuoka, A., 127–130 matrix. See also comatrix – (h – h)– – of Hopf endomorphism of Symm, 398 – bideterminant, 6 – Cartan, 475 – – elliptic, 556 – – generalized, 517 – – graph of symmetric, 514 – – indecomposable, 516–517 – – of quiver, 532 – – symmetrizable, 514–515

Index

– to conjugation, 522 – C-rings, 308 – DE – – of SymmR , 399–400 – determinant and, 5–7 – domination of, 14 – dual invariant, 18 – generalized diagonal of, 17 – GL n primitive, 155 – of Hopf algebra endomorphism of Symm, 398 – line of, 17 – monomial, 9 – nonsingular, 8–9 – permanent, 7 – of quadratic form, 38 – rank functions and, 14–15 – ring contexts, 307–308 – semiring theory and, 5–6 – sign-nonsingular, 6 – singularity for, 7–10 – – right/left, 7 – symmetrizable, 514 – tropically singular, 9–10 matrix union, 22, 29–31 max-algebra, 5, 21–22 Merkurjev, A., 37, 44–47, 54 Merkurjev theorem, 56–57 Mestre, J. F., 48 Militaru, G., 182 Milnor – conjecture, 45–47, 61 – K theory, 44–45 – Witt rings theorem of, 63 Milnor, J., 36, 44–45, 63. See also Knebusch-Milnor exact sequence min-algebra, 5 Minkowski, H., 37, 69 Minoux, M., 11, 20 Miwa, T., 489 mixed distributive law, 182 – entwined structures and, 251 Möbius function, 362, 395 modules with ﬂat connections, 293 monads, 242–243, 418–419 – adjoint functors and, 243 – algebra over, 243 – of descent, 280 – formal theory of, 310 – module over, 243

Index

monoid – multiplication of, 176 – – as category, 240 – product of, 176 – unit of, 176 monoidal – anti-, 197 – autonomous, 232 – bialgebroids, 231 – bicategory, 186 – bimodules, 177 – braided category, 196 – category, 98, 242–243 – closed structure, 101 – comodules, 287–288 – forgetful functor, 199 – functors, 244 – – B|A-corings and, 253 – – corings and representable, 253 – K-modules, 176 – Morita equivalence, 195 – Morita theory, 194–195 – morphisms, 185 – multiplication of, 244 – multiplication of monoid in, 176 – product of monoid in, 176 – R-bimodules, 196–197 – right closed, 231 – right module, 177 – unit of, 244 – unit of monoid in, 176 – weak center, 202 monomial realization, 479 monomorphism, 240 – products and, 265 Moore complex, 116 Moreau, C., 428 Moresi, R., 53 √ Morita. See also Morita-equivalent – base change, 191–192 – contexts, 281–282 – – category of, 282 – – comodule and, 282–283 – – for coring extension, 285–286 – – Glaois comodules and, 286 – – strict, 282 – equivalent, 282 – – monoidality of, 195 – monoidal theory, 194–195 Morita-Takeuchi context, 308 morphisms. See also isomorphisms

601

– Bernstein, 439–440 – C-ferential, 149 – comodules and factorizable, 256 – corings and, 245 – – algebra of coinvariants of, 260–261 – – coinvariants of, 260–261 – equivalent – – of covers/noncommutative spaces, 299 – ghost component, 446 ˆ – – of Burnside ring B(Z), 446 – identity, 239 – irreducible, 528 – of λ-rings, 411, 423 – pure, 264 – reﬁnement, 299 – Symm and comultiplication, 367–368 – types of, 240 Moscovici, H., 224 Muirhead inequalities, 440 multiple composition – cubical map and, 106 – HHvKT and, 105 multiplication, 176 – on Abelian groups (A), 350–351 – balanced, 198 – functor as, 244 – of monoid, 176 – – as category, 240 – of monoidal category, 244 – on NSymm, 377 – p-adic Witt polynomial, 333–334 – p-adic Witt polynomials, 333 – p-adic Witt vectors and p, 336 – on Symm, 377

N Nagata, M., 343 Nakajima, H., 479, 510, 549 Nakashima, T., 474, 479, 491 natural transformations, 241 necklace – algebra, 429–430 – deﬁnition of, 428 – polynomials, 427–428 – – formulas for, 428–429 – rings, 427 – – deﬁnition of, 430 – – Frobenius, 431–432 – – modiﬁed, 432–433 – – Verschiebung, 431–432

602

– Witt vector v., 432 negative root, 514, 516 nests, 393 – Witt vectors and, 394–395 – – iterated, 398 nilradical, 41–42 NSymm – cofree coalgebra and, 381 – curves, 370–371 – multiplication on, 377 – QSymm duality between, 373–374 – structure of, 369 – universal property of, 370 noncommutative schemes, 299 non-formally real ﬁelds, 39 nonsingular, 8 – quadratic forms, 38 normalized integral, 270 norm residue symbol, 45

O objects, 239 Ojanguren, M., 62 Olum, P., 115 opposite category, 239 orbit category, 551 orbit groupoids, 89 orderings, 64 oriented cycle, 520 Orlov, D., 46 orthogonal – complement, 61 – direct sum, 38, 61 Osiak, K., 41 overlapping shufﬂes, 372

P p-adic valuation, 329 p-adic Witt polynomials, 331–332 – addition/multiplication, 333 p-adic Witt vectors – ﬁeld of characteristic p and, 342 – Frobenius operations and, 335–336 – functor construction for, 331–333 – historical motivation of, 329–331 – multiplication by p for, 336 – over ring p, 338, 340–341 – rigidity of, 341 – ring units of, 340 – Teichmüller representative and, 334–335 – topology of, 334

Index

– uniqueness of, 341 – Verschiebung operation for, 335 Pang, C. K., 58 Pardon, 62 Pareigis, B., 218 Parimala, R., 45 Parshall, B., 510 partial boxes, 107–108 – collapsing, 108 partitions, 353–355 – conjugate, 440–441, 497 – even, 495 – generalized, 484 – length of, 483 – weight of, 353 path. See also Littelmann path model – algebra – – of quivers, 525–528 – – quotient of, 509 – length of, 519 – nontrivial, 519 – in quiver, 519 – realizations, 479 PBW basis – for C () of ﬁnite type of quivers, 540–541 – for C () of tame type of quivers, 549–550 Peiffer, R., 88 Peng, L., 510, 551, 555, 557–558 permutation, cycle type, 435 Pﬁster. See also Arason-Pﬁster Hauptsatz – forms, 38–39 – level theorem, 58 – local-global principle, 42 – theorem, 60 Pﬁster, A., 40, 63 Picard group for corings, 310 Picard-Vessiot (PV) theory – basics, 128–130 – characterization, 168–170 Picken, R. F., 118 plactic algebra, 491 planar – orientation, 567 – rotations and reversals, 575–576 plethysm – calculating, 425–426 – deﬁnition of, 424–425 – inner/outer, 425, 441 – – problem of, 442 – operations – – distributivity of, 426–427

Index

polynomials, 70. See also p-adic Witt polynomials – Hall, 537 – Hirzebruch, 407–409 – necklace, 427–428 – – formulas for, 428–429 – rings – – Hermetian/quadratic forms over, 37 – Witt, 325–326, 451–452 – – addition/multiplication, 333 – – p-adic, addition/multiplication, 333–334 Ponomarev, V. A., 531. See also Bernstein-Gelfand-Ponomarev (BGP) reﬂective functors Porchet, Y., 60 Porter, T., 116, 118–119 positive root, 514, 516 positive selfadjoint Hopf (PSH) algebras, 438–439 power operations, 414 power series – in inﬁnitely many variables, 452–453 – quasisymmetric, 453–454 – symmetric, 453–454 P-module, 87 preadditive category, 239 preinjective indecomposable module, 543 preorderings, 68 preprojective indecomposable module, 543 Pressley, A., 549 prime ideals, 41 principal bundles – coalgebra, 295 – free actions and, 293–294 – as Hopf-Galois extensions, 294–295 – noncommutative, 293 principal extensions, 296–297 product. See also coproduct; cotensor products; inner product; smash product; Takeuchi product; tensor product – category, 239 – comultiplication, 366–367 – – inner product v., 368 – crossed, 224 – – central simple algebras and, 347–348 – formula, 337–338 – monomorphism and, 265 – overlapping shufﬂe, 372 – in A-ring, 252 – Sweedler – – of Jacobson-Bourbaki correspondence, 278

603

projection sectioning, 395–396 PV. See also Liouville; L/K extension; Picard-Vessiot (PV) theory – extension, 128 – – C-ferential, 152–153 – – characterization of, 141–142, 155 – – of AS D-module algebras, 164–168 – – equivalence, 148 – – formalism of, 130–135 – group scheme, 128 – – of L/K extension, 137–138 – Hopf algebra, 128–129, 153, 164 – – of L/K extension, 134 Pythagoras number, 60–61 Pythagorean ﬁelds, 42

Q QSymm, 369 – cofree coalgebra and, 381 – comultiplication on, 374–375 – – co-unit for, 376–377 – Hopf algebra structure on, 373 – λ-ring structure on, 422 – NSymm duality between, 373–374 – primitives of, 374 – of quasi-symmetric functions, 371 quadratic – forms – – anisotropic, 38–39 – – classiﬁcation of, 47 – – classiﬁcation/representation problem in theory of, 39 – – determinant of, 38 – – diagnolized, 38 – – dimension, 37 – – discriminant of, 40 – – equivalence of, 37, 47–49 – – equivalent, 37–38 – – Euler, 531 – – function ﬁelds and, 55–56 – – general classiﬁcation theorem of, 48–49 – – generic splitting towers of, 55 – – Hasse invariants of, 43–44 – – Hauptsatz on, 40 – – hyperbolic, 39 – – invariants of, 47 – – isomorphisms and, 46 – – isotropic, 38–39 – – literature on, 37 – – matrix of, 38 – – nonsingular, 38

604

– – origins of, 36 – – over afﬁne algebra, 37 – – over general rings, 37 – – over global ﬁelds, Hasse principle, 51 – – over number ﬁeld, 48 – – over polynomial rings, 37 – – reduced theory of, 67–68 – – similar, 39 – – system of, 57 – – theory of, 69–70 – – total signature and, 47 – – as trace forms, 48 – – universal, 56 – – Voevodsky’s theorem for, 47–48 – – Witt classes of, 39 – – Witt equivalent and, 39, 49–50 – – Witt invariants of, 44–45 – space – – determinant of, 38 – – isometric, 37 quantum – group, 474–476 – – canonical bases of, 510 – – deﬁnition of, 518 – – Kac-Moody algebra and, 514–519 – – Ringel-Hall algebras and, 538–539 – – root vectors in, 540 – groupoids, 206 – torsors, 200–202 quantum ﬁeld thoery, 274 – Hopf algebra and symmetry, 174 quantum group Uq (gln ), 476 quasi-coherent sheaves, 297–298 quasi-ﬁnite injector, 308 quasi-symmetric functions – monomial, 371–372 – QSymm of, 371 quaternion algebra, 44 Quillen’s K-theory, 45 quivers, 509 – Auslander-Reiten, 529–530, 534–535, 546 – Cartan matrix of, 532 – connected, 519 – ﬁnite type of, 524 – – PBW basis for C () of, 540–541 – Kronecker, 523, 545–549 – modulated, 527 – path algebra of, 525–528 – path in, 519 – representation of – – category/direct sum/indecomposable, 520

Index

– – dimension vector/simple, 521 – representations and, 519–522 – – category of, 520 – resources for, 559 – tame type of, 524 – – PBW basis for C () of, 549–550 – – root vectors in C () of, 541–545 – variety, 510 – wild type of, 524

R rank functions – arithmetic behavior of, 22 – column, 15–16 – – bounds and, 25 – – spanning, maximal, enveloping, 16, 19 – combinatorial, 17 – factor, 15 – Kapranov, 18 – matrix and, 14–15 – of matrix union, 22, 29–31 – relations between, 18–22 – row, 15 – – bounds and, 25 – – spanning, maximal, enveloping, 16, 19 – semiring and, 14–15 – symmetric, 17 – tropical, 17 rank-sum inequalities, 22–26 rational element, 266 rational functions, 70 rational submodule, 266 R-bimodules, 193–194 – monoidal, 196–197 R-corings, 175 – co-opposite of, 179 – cosemisimple, 217 – coseparable, 217 – Galois, 180–181 – – duality, 181–182 – grouplike element in, 180 – left comodule morphism of, 180 – morphism of, 179, 185 – right comodule morphism of, 179 realizable sequence of integers, 362 reciprocity formula, 429 recording tableau, 489 reﬁnement, 299 regular indecomposable module, 543 regular invariant, 567 Reidemeister, K., 88

Index

Reiten, I., 509 relative Hopf modules, 213 representations – of ﬁnite-dimensional algebras, 509 – quivers and, 519–522 – – category of, 520 representation theorem, 68 residue symbol, 45 Retakh, V. S., 370 Reutenauer, C., 359 Reznick, B., 60 right. See also left-right symmetry – adjoint, 241 – almost split, 528 – basis, 12–13 – bialgebroids, 183–186 – character – – on R-rings, 178–179 – closed – – monoidal, 231 – coaction, 254 – cointegral, 215 – comodule algebra, 212–213 – comodule morphism – – of R-corings, 179 – comodules, 254 – dual of R-rings, 181 – H-comodule, 209 – integral, 215 – invertible, 9 – linear combination, 10 – linearly independent, 11 – linear span, 10–11 – module – – ﬁrm, 270 – – monoidal, 177 – – for R-rings, 178 – semimodules, 10, 12 – singular, 8 – – matrix, 7 Riley, K. F., 566 Ringel, C.-M., 509, 511, 539–540, 547, 551, 554, 558 Ringel-Hall algebras, 510, 537–538 – quantum groups and, 538–539 – untwisted, 553–554 rings, 4. See also Ae -rings; β-rings; Burnside ring; CRing; C-rings; Dedekind, rings; L-rings; λ-rings; R-corings; R ⊗k Rop -rings; R(S) ring; R-rings; special lambda ring; σ-rings; -rings

605

– algebra and, 252 – aperiodic, 434 – associative commutative, 9 – binomial – – adjoints of, 433 – – deﬁnition of, 430 – – free, 430–431 – – λ-rings v., 431 – characteristic p, perfect, 338 – Cohen, 342–343 – – of k, 344 – contexts – – matrix, 307–308 – differential, 130, 143 – – L/K extension and principal, 132–133 – – simple, 136 – – subring of constants of, 130 – dual – – of comatrix coring, 253 – – comodules and, 265 – – natural endomorphism, 252 – – of Sweedler coring, 253 – Frobenius, 218–220 – Galois extensions and noncommutative, 300–301 – Grothendieck, 445 – Hermetian/quadratic forms over, 37 – of Hopf endomorphism – – of Symm, 392–393 – lambda, 328 – level of, 36, 38 – necklace, 427 – – deﬁnition of, 430 – – Frobenius, 431–432 – – modiﬁed, 432–433 – – Verschiebung, 431–432 – nonunital – – strong structure theorem over, 281 – p-adic Witt vectors – – units of, 340 – p-adic Witt vectors over p, 338, 340–341 – polynomial – – Hermetian/quadratic forms over, 37 – R(S), 436 – sigma, 412 – total quotient, 163 – valuation, 339 – Witt, 329 – – Kula’s theorem on, 66–67 – – Milnor theorem of, 63 – – nilradical of, 42

606

– – of semilocal ring, 62 – – zero divisors in, 42 – Witt rings of commutative, 61–62 β-rings, 451 Robinson-Schensted correspondence, 486, 488–489 – crystal version of, 489 Robinson-Schensted-Knuth correspondence, 491–495 – dual-, 495–497 Roiter, A., 238 root – category, 551–552 – imaginary, 516 – negative/positive, 516 – regular/irregular, 544 – simple, 516 – system, 516 – vectors, 540 – – in C () of tame type of quivers, 541–545 – – imaginary, 547 Rosenberg, A., 62, 65, 315 row rank, 15 – bounds and, 25 – spanning, maximal, enveloping, 16, 19 R-corings – right/left dual of, 181 R(S) ring, 436 R ⊗k Rop -rings – source/target map of, 183 R-matrix, 196 R(S) ring, 434–435 – Hopf algebra structure on, 437 – structure on, 437–438 R-rings, 175 – Galois, 179 – – duality, 181–182 – K-algebra and, 177–178 – left character on, 178 – left module morphism for, 178 – morphism of, 177 – right character on, 178–179 – right dual of, 181 – right module morphism for, 178 R-S torsor, 201–202 R-S torsor – faithfully ﬂat, 201 Ruch-Schönhofer theorem, 441

Index

S S3 – regular invariants of, 571–576 – – canonical projections for, 572–573 – – interpreting, 573–574 – – set of bases for, 574–576 S4 – regular invariants of, 576–582 – – in dimension of 2, 577–580 – – in dimension of 3, 580–582 Saito, K., 555–556 scalar – extension, 193, 202, 207 – multiplication, 10 – product, 435 Scharf, T., 359 Scharlau, W., 37, 48, 67, 71 Schauenburg, P., 187–188, 191, 193, 200, 224, 230–231 Schensted column bumping algorithm, 487 Schensted-Knuth correspondence, 474 Schmid, H. L., 330 Schmidt, F. C., 342. See also Krull-Remak-Schmidt theorem Schmidt, F. K., 330 Schneider, H.-J., 224, 227 Schoeller, C., 343 Schreier, O., 70 Schur – algebra, 510 – functions, 355–356 – – skew, 364 – positive, 359 Schur, I., 434, 436 Schützenberger, M., 491 Scott, L., 510 semigroup with devisor theory, 53 semi-inﬁnite words, 498 semilocal rings, 62 semimodules – left basis of, 12 – left/right, 10 – linear dependence/independence and, 11 – right basis of, 12 semiring – antinegative, 4, 8, 12 – – bounds and, 25–26, 28–29 – arbitrary – – bounds and, 24, 27

Index

– Boolean, 5 – – bounds and, 23, 26–28 – chain, 5 – commutative, 4 – matrix theory on, 5–6 – rank functions and, 14–15 – zero-sum-free, 4 separable – bimodules, 270 – – comatrix corings and, 271 – C-rings, 306 – functors, 271 Serre, J.-P., 48, 570 Serre relations, 556 shape λ, 498–499 Shapiro, D. B., 59 Sharko, V. V., 119 sheaf – coherent, 511 – perverse, 510 – quasi-coherent, 297–298 shufﬂes, 372 Siebeneicher, C., 443 sigma ring, 412 simple reﬂection, 516 simple root, 514, 516 simplicial – groups, 116–117 – spaces, 114 – T -complex, 117–118 Singer, M. F., 128 singularity, 7–10, 30–31 sink sequence, 536 skew ﬁelds. See ﬁelds Sladek, A., 52, 64, 67 smash product, 198 Snapper Liebler-Vitale Lam theorem, 441 Snapper ordering, 440 Solomon, L., 377 spaces. See also Fock space representations; G-spaces; subspace – bilinear, 37 – Boolean, 41, 68 – classifying, 113–114 – – BU, 327 – – simplicial, 114 – cover, 299 – cubical classifying, 101–102 – – ﬁltered, 99

607

– Fermionic Fock, 498 – ﬁltered, 98 – – connected, 103 – as Grothendieck category, 297 – inner product – – metabolic, 61 – – Witt equivalence and, 61–62 – integral, 298 – noncommutative – – comodules as, 297–298 – – equivalent morphisms of, 299 – – maps between, 297 special lambda ring, 410 Spencer, C., 93 split – C-rings, 306 – extension, 270 – left almost, 528 – morphism, 240 – right almost, 528 Springer, T. A., 64, 66, 71 Springer’s theorem – of Witt groups, 66 – of Witt rings, 64 squares – classes of, 37 – – Witt equivalent, ﬁelds and, 50–51 – class numbers, 50 – groupoids and commutative, 91–92 – horizontal composition, 94 – sums of, 36, 58 – – composition for, 59 – – level and, 58–59 – – Pythagoreas number and, 60–61 – vertical composition, 94 Sridharan, R., 45 σ-rings – λ-rings v., 411–412 – structure of, 412–413 stable tube, 545 stalk complex, 551 Stembridge, J. R, 485 Street, R., 186 Strehl, V., 429 Strong Structure theorem, 227 subcategory, 239 – rigid, 156 subdivision – algebraic inverse to, 90, 94, 101, 106

608

subgenerated modules, 267 subspace, weakly closed, 298 Sugano, K., 216 sum of squares, 36, 58 supernatural – numbers, 394 – quotients – – of big Witt vectors, 393–394 S|R-corings – morphism of, 185 Sweedler – coring, 246 – – dual ring of, 253 – C-rings, dual-, 306–307 – index, 176, 184, 191, 205 – product of Jaconson-Bourbaki correspondence, 278 Sweedler, M. E., 135, 153, 218, 238 Sweedler-Heyneman notation – coassociativity of, 245 – comodules and, 255 Sylvester’s laws, 22, 26–29 Symm, 349. See also NSymm; QSymm – autoduality of, 374 – for big Witt vectors, 363 – cofree coalgebra and, 381 – comultiplication morphism on, 367–368 – forms of, 327–328 – free algebra and, 378 – Frobenius operations and, 383–386 – grading, 364–365 – homogenous basis for, 358 – Hopf algebra endomorphisms of – – matrix of, 398 – Hopf endomorphisms of – – ring of, 392–393 – inner product on, 355 – Liulevicius theorem and, 365–366 – λ-ring structure on, 421–422, 425 – (h – h)-matrix of Hopf endomorphism of, 398 – multiplication on, 377 – primitives of, 365 – projective limit of, 454 – unary operations on, 422–423 – Verschiebung operation and, 382 symmetric function. See also plethysm; quasi-symmetric functions – algebra of, 452–454 – forgotten, 357 – monomial – – calculating, 356

Index

– – partitions and, 353–355 – power sum, 357–359 – Schur, 355–356 – theory, 327, 353 symmetric powers, 448–449. See also power series symmetric rank, 17 symmetrizable matrix, 514 symmetry studies, 567–571 SymmR , 388–389 – autoduality of, 399 – curve in, 398 – DE matrix of, 399–400 syzygies, 88 – chain of, 101 Szczepanik, L., 50 Szlachányi, K., 183, 185, 188, 194, 219, 232

T

T(C+ ) – bialgebra, 154–156 – Birkhoff-Witt, 158 tableaux realizations, 479 Takeuchi, M., 128, 133, 182–184, 221, 224, 238. See also Morita-Takeuchi context Takeuchi product, 183–184, 189, 287 Tannaka category, 130 Taylor, J., 89 Teichmüller, O., 325–326, 330, 343 Teichmüller representative, 331, 342 – p-adic Witt vectors and, 334–335 – in Wp∞ (A), 334 tensor coalgebra, 378 tensor module algebra, 150 tensor product, 38. See also cotensor products – of crossed complexes, 111–112 – of crystal base, 478–479 – of modules, 308–309 – rule, 479 Thibon, J.-Y., 359, 370 thin element, 97, 102, 118 – algebraically, 108 – geometrically, 108 three-shell. See cubes Tignol, J. P., 57, 64 T isomorphism, 446–448 Tits form, 532 Todd genus operation, 409 topology – of p-adic Witt vectors, 334 – splitting principle in, 401

Index

torsion, 42 torsor, 201 total signature, 47 trace forms, 48 translation function, 294 triangulated category, 551 – Witt groups of, 64 tropical – algebra, 5 – rank, 17 – singularity, 9–10, 30–31 Tsen theorem, 60 tubular extension, 557 twist, 189. See also cocycle double twist; Drinfel’d twist twisted convolution algebra, 293 2-cocycle, 189 – invertible, 223

U U-injective, 226 u-invariant, 56–57 unique existence theorem, 143–144 unital C-action, 149 units, 176, 252 universal – differential form, 259 – lower bound, 5 – upper bound, 5 upper bound, 5 Uq (gln ) – Hopf algebra on quantum group, 476 – – module – – integrable, 475

V Van Kampen, E. H., 89 van Kampen theorem (vKT), 85, 90. See also higher homotopy van Kampen theorem (HHvKT) – approaches to, 90–92 Verschiebung ˆ – on Burnside ring B(Z), 449 – endomorphism – – Frobenius endomorphism compared to, 391 – necklace rings, 431–432 – operation, 449 – – ghost component characterization of, 382 – – for p-adic Witt vectors, 335 – – Symm and, 382

609

vertical identities, 94 vertical subcategory, 186 Viana, M., 566–567, 570, 577 Vishik, A., 46, 54, 57 Vishik gap theorem, 54–55 Voevodsky, V., 36, 46–47 – quadratic form theorem of, 47–48 voting preferences, 575

W Wadsworth, A. R., 45, 57 Wall, C. T. C., 112 Ware, R., 62, 65 weakly closed subspace, 298 Weierstrass, K., 523 weighted projective lines, 511 Wensley, C. D., 89, 113 Weyl group, 516 w-groupoids (w-Gpd), 113–114 – categories of, 102–103 – with connections, 98 – cubical, 101–102 – – ρX∗ , 104–106 – globular, 117 Whitehead, J. H. C., 86–88, 106, 112, 113–115 Whitehead theorem, 104 Wilkerson theorem, 421–422 Wisbauer σ-category, 267 Witt. See also p-adic Witt polynomials – algebra, 329 – classes – – dimension-index and discriminant for, 40 – – of quadratic forms, 39 – equivalent – – ﬁeld square class number and, 50–51 – – of global ﬁelds, 51–53 – – inner product spaces and, 61–62 – – quadratic forms over, 39, 49–50 – groups, 50–51, 329 – – Knebusch, 64 – – Springer’s theorem of, 66 – – of triangulated gategory, 64 – index, 55 – invariants, 44–45 – polynomials, 325–326, 451–452 – – addition/multiplication, 333 – – integrality theorem of, 359–360 – – p-adic, addition/multiplication, 333–334 – rings, 329 – – abstract, 65–67

610

– – arbitrary, 64 – – of bilinear forms, 39–40 – – commutative rings and, 61–62 – – of Dedekind domains, 63–64 – – elementary type of, 66 – – fundamental ideal, 40 – – of global ﬁelds, 52 – – Kula’s theorem on, 66–67 – – Milnor theorem of, 63 – – nilradical of, 42 – – prime ideals, 41 – – Pythagorean ﬁelds and, 42 – – reduced, 68 – – of semilocal ring, 62 – – skew ﬁelds and, 64 – – Springer’s theorem of, 64 – – torsion, 42 – – units of, 42–43 – – zero divisor in, 42 – terminology, 329 Witt, E., 36, 62, 70–71, 330, 344, 353, 382, 428 Witt-Burnside functors, 450–451 Witt vector – of formal group, 361 – ghost, 406 – universality of property of, 340 Witt vectors. See also p-adic Witt vectors; Wp vectors – adding disjoint, 337 – adding homothety operators on, 327 – big/large/generalized, 325 – carryless, 433 – cofree, 420–421 – comonad structure on, 419–420 – coordinates, 358–359 – disjoint, 337 – as endomorphism, 402–404 – ﬁnite length, 325 – ﬁxed point indices of iterated maps and, 362–363 – functor of big, 348–349 – history of, 325–327 – information sources on, 328–329 – λ-rings and, 415 – necklaces v., 432

Index

– nested, 394–395 – – iterated, 398 – product formula for, 337–338 – p-th power in, 336 – supernatural quotients of big, 393–394 – Symm for big, 363 – universal property of, 371, 409–410 Wp∞ (A) – p-th powers in, 336 – Teichmüller representative in, 334 Wp∞ vectors, 404–405 Wronskian, 131 – criteria, 151 – relations, 354

X Xiao, J., 510, 540, 551 Xu, P., 182

Y Yang-Baxter equation, 174, 193 Yetter-Drinfeld modules, 191–193, 202–203, 290–291 – anti-, 291 – right-right, 202–203 Yoneda lemma, 138 Yoshii, D., 555–556 Young, A., 434 Young diagram, 483 – generalized, 484 – weight of, 483 Young tableaux, 474 – crystal of, 482–484 – – rational, 484–486 – shape of, 483 – standard/semistandard, 483

Z Z(p) -algebras, 396–398 Zelevinsky classiﬁcation theorem, 439 Zelevinsky structure theorem, 437 Zelvinsky, A. V., 438 zero divisors, 4 Zhang, G., 549 Zorn lemma, 68